STATISTICS AND PRINTING:
APPLICATIONS OF SPC AND DOE
TO THE
WEB OFFSET PRINTING INDUSTRY
A Project
Presented
to the Faculty of
California State University, Dominguez Hills
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
in
Quality Assurance
by
Craig P. Paxson
Fall 1993
Copyright by
CRAIG P. PAXSON
December, 1993
All Rights Reserved
PROJECT: STATISTICS AND PRINTING: APPLICATIONS OF
SPC AND DOE TO THE WEB OFFSET PRINTING
INDUSTRY
AUTHOR: CRAIG P. PAXSON
APPROVED:
E. Eugene Watson, Ph.D.
Project Committee Chair
Milena Krasich, P.E.
Committee Member
William Trappen, P.E.
Committee Member
iv
TABLE OF CONTENTS
Page
Chapter 1: Introduction...................................11
TQM/SPC in Printing.............................11
SPC.............................................12
DOE.............................................12
Process Capability..............................13
Measurement Science.............................13
Chapter 2: Statistical Process Control....................15
Fundamentals of SPC.............................15
Basic Theory of Control Charts ...............15
Terminology................................16
Types of Control Charts....................17
Rational Subgrouping.......................17
Steps in Implementing SPC..................18
Setting up a Control Chart.................19
Reacting to Out of Control Conditions......19
Shewhart Control Charts.........................20
Pattern Analysis...........................20
Variables Control Charts .....................26
Introduction...............................26
X-bar and R Charts.........................26
Individuals Charts.........................34
Attributes Control Charts ....................36
Introduction...............................36
Sampling Plans.............................37
v
Fraction Nonconforming - p-, np- Charts....37
Nonconformities - u-, c- Charts............42
Advanced Control Charts.........................46
Short-Run SPC ................................46
Data Normalization ...........................47
Nominal....................................47
Target.....................................48
Short Run..................................48
X-bar and R Charts ...........................49
PRE-Control ..................................55
Introduction...............................55
Construction...............................55
Use of PRE-Control.........................57
Potential Applications.....................58
Conclusion......................................59
Chapter 3: Design of Experiments..........................61
Introduction....................................61
Fundamentals of DOE.............................61
Terminology................................61
Steps in Design of Experiments.............62
Statistical Analysis .........................65
One Way ANOVA..............................65
Two Way ANOVA..............................66
Graphical Analysis ...........................67
Main Effect Plots..........................68
Interaction Plots..........................69
vi
One Factor Experiments..........................70
Factorial Experiments...........................72
The Taguchi Methods.............................73
Introduction...............................73
Limitations................................75
Orthogonal Arrays..........................75
Linear Graphs..............................76
Assigning Factors..........................77
Analysis...................................79
L12 No Interactions........................81
S/N Ratios.................................82
Parameter Design...........................83
Conclusion......................................87
Chapter 4: Process Capability and Measurement.............88
Introduction....................................88
Process Capability..............................88
Definitions................................89
Process Capability Using a Histogram.......89
Estimating Natural Process Limits..........90
Process Capability Ratios..................91
Potential Applications.....................92
Measurement Science.............................93
Definitions................................93
Gage Repeatability and Reproducibility.....94
Conclusion......................................99
Chapter 5: Conclusion....................................100
vii
Appendix 1 - Tables for Control Limits...................101
Appendix 2 - Taguchi Experiment Tables...................103
Appendix 3 - GR&R Forms..................................106
Bibliography.............................................108
viii
LIST OF TABLES
Table Page
Table 1. X-bar and R Chart Example Data...............32
Table 2. Individuals Chart Example Data...............35
Table 3. p-chart data.................................39
Table 4. Holes/Plate Example Data.....................44
Table 5. Nominal Short Run Example Data...............50
Table 6. Short Run Example Data.......................53
Table 7. One-Way ANOVA Table..........................65
Table 8. Two-Way ANOVA Table..........................67
Table 9. One Factor Experiment Example Data...........71
Table 10. One Factor Example ANOVA Table..............72
Table 11. L4 Array....................................76
Table 12. Interaction Table...........................76
Table 13. Sum of Squares example table................80
Table 14. L8 ANOVA Table example......................81
Table 15. Parameter Design Arrays.....................85
Table 16. Parameter Design Example Data...............86
ix
LIST OF FIGURES
Figure Page
Figure 1. Out of Control Condition #1.................21
Figure 2. Out of Control Condition #2.................22
Figure 3. Out of Control Condition #3.................23
Figure 4. Out of Control Condition #4.................24
Figure 5. Non-Random Pattern..........................25
Figure 6. X-bar and R Chart Example...................33
Figure 7. Individuals Chart Example...................36
Figure 8. Plate Defect p-chart........................40
Figure 9. Plate Defect np-chart.......................42
Figure 10. Holes/Plate Example u-chart................45
Figure 11. Nominal Example X-bar chart................51
Figure 12. Short Run X-bar and R Chart................54
Figure 13. PRE-Control Chart..........................56
Figure 14. Example PRE-Control Chart..................59
Figure 15. Main Effect Plot...........................68
Figure 16. Interaction Plot...........................69
Figure 17. Interaction Plot...........................69
ABSTRACT
The printing industry has recently had an explosion of
interest in Total Quality Management (TQM) and Statistical
Process Control (SPC). Printers are beginning to realize
the positive effects that statistical tools such as SPC can
have on quality and productivity. This paper will present
ideas on applying the statistical tools of SPC and Design of
Experiments (DOE) to the web offset printing industry.
The paper is not intended as a primer on the theory or
mathematics involved in the tools; rather it is a treatise
on applying SPC and DOE to the printing industry.
Hopefully, the reader will garner new ideas for statistics
in printing, or be excited at the possibilities of using SPC
and DOE in the industry.
11
CHAPTER 1: INTRODUCTION
TQM/SPC in Printing
The printing industry has experienced a Renaissance in
the last several years in regards to quality. This growth
in interest was spurred on by the intense competition in the
industry, and included a growth in interest in Statistical
Process Control (SPC). Many printing companies have decided
to implement Statistical Process Control, especially
Shewhart Control Charts. Unfortunately, in many cases the
technical knowledge of statistics was not present, leading
to control charts that were invalid or misleading.
This project will present statistically correct methods
of using SPC and Design of Experiments (DOE) in the web
offset portion of the printing industry. Potential
applications, examples, and proper use of the tools of SPC
and DOE will be presented. The paper will be beneficial to
anyone in the industry who is interested in increasing
quality and productivity, and decreasing costs.
This paper will not thoroughly cover statistics, rather
it will show how to use statistics, SPC and DOE for gain in
the printing industry. Several good primers for statistics,
SPC and DOE are listed in the bibliography.
12
SPC
Statistical Process Control will be covered from an
applications standpoint. Basic introduction to the control
chart, including basic theory, subgrouping and pattern
recognition will be presented, followed by examples and
application for the major control charts.
The term SPC has become almost a non-word in the last
several years. The addition of other tools besides the
control chart, and in some cases a certain style of
management, has made the term SPC different to many. In
this paper, SPC will be addressed as control charts only.
Other problem solving tools, such as Pareto analysis, cause
and effect diagrams, etc. and styles of management, such as
continuous improvement and TQM, will not be covered as a
part of paper. Control charts are a tool that can be used
with any style of management, although their effects may be
greatly enhanced when combined with other problem solving
tools and proper management.
Further study and in-depth discussions of control
charts may be found in any one of the books listed in the
bibliography.
DOE
The next chapter describes Design of Experiments, a
relatively new field in printing. Both the classical and
13
Taguchi design will be presented, again with the emphasis on
applications of DOE to printing. The Taguchi method, no
matter how controversial, is easily implemented and
understood, and will be presented as the main experimental
tool. The use of inner and outer arrays has great
application to printing, where many factors cannot be
readily controlled.
The theoretical discussion of DOE, and texts on the
statistical methods (ANOVA, etc.) may be found in the books
listed in the bibliography.
Process Capability
Process Capability will be addressed in the last
chapter. An understanding of the tools presented in the
first two sections should enable the reader to use this
format to improve and maintain his process. Process
capability enables printers to measure how changes in the
process compare to internal or external specifications.
Measurement Science
SPC and DOE are based on measurements, and for these
tools to be useful the measurements must be valid.
Measurement Science deals with the process of taking
measurements so they are accurate and repeatable. The last
14
chapter will show easy ways of conducting Gage Repeatability
and Reproducibility (GR&R) studies, the main tool of
measurement science.
15
CHAPTER 2: STATISTICAL PROCESS CONTROL
Fundamentals of SPC
The term Statistical Process Control can be defined by
defining its three component words. The word statistical
means "having to do with numbers" or "drawing conclusions
from numbers." A process is any system of causes, a
combination of conditions which create some output. Control
means "to make something behave the way we want it to behave
(AT&T, 1984)."
Putting these three terms together, and applying it to
printing, we find that statistical process control means
that with the help of numbers, we study our process in order
to make it behave the way we want it to behave (AT&T, 1984).
It is this use of numbers that is the key to
statistical process control. Making changes to the process
is not difficult - but it is the statistics that tell us
when to make a change, or how much of an effect our changes
are having on the process.
Basic Theory of Control Charts
The fundamental theory of control charts is based on
the fact that everything varies, but varies in some way that
is predictable. For instance, solid ink density varies, but
it has some point it varies around, and some amount of
16
spread from that point. Statistics can provide us with the
knowledge of that point, how much spread there is, and the
chances that the density will stray from that point.
Terminology
Chance and Assignable Causes. The factors that affect
the set point and the spread are divided into two groups -
chance and assignable causes. These terms were coined by
Walter Shewhart (the inventor of control charts), and
modified by W. Edwards Deming.
Chance causes are those causes that randomly appear and
make the process behave in some random, unpredictable
pattern. Such "chance" variation is relatively stable over
time, because it is the result of many contributing factors.
Assignable causes are those causes that sporadically
appear and have impact on the process. These factors are
identifiable, and can be assigned the deviations that they
cause.
Statistical Process Control charts help us to
differentiate between these "chance" and "assignable" causes
of variation by using charts that show what the chance
variation should look like. Once we have identified
variation that is not caused by chance causes, we can learn
what caused the variation, and eliminate it from the
process, thus improving the process.
17
Attribute. Attribute data is characterized by counts,
such as number or percentage defective, number of defects,
etc.
Variable. Variable data is characterized by
measurements on a continuous scale, such as length or
weight.
Subgroup. A sample of more than one individual from a
process is called a subgroup.
Control limits. Control limits are lines drawn on a
control chart that define a band of allowable variation in
the process.
Statistical Control. A process is in a state of
statistical control if it exhibits only random variation.
Types of Control Charts
Control charts can be divided into two main classes,
attributes and variables control charts. Attributes control
charts are based on definite numbers, such as percentages or
counts. They include p-, np-, c- and u-charts. Variables
control charts, such as X-bar and R, are based on
measurements.
Rational Subgrouping
One of the most important concepts in SPC is the
concept of subgrouping. Since most of the charts are based
on statistics from some set of numbers, how those numbers
18
were arrived at is extremely important. This concept is
known as subgrouping, and properly chosen subgroups are said
to be rational subgroups.
Proper determination of sample sizes varies for each
type of control chart and will be discussed further in each
section.
Steps in Implementing SPC
Statistical Process Control is a major part of any
quality program, including Total Quality Management.
Because of its complexity, steps must be taken to ensure it
is implemented correctly. Some potential problems that must
be addressed are the measuring system, who will plot the
chart and calculate the control limits and who will
interpret the chart. Training these personnel must be a key
part in any SPC program.
Generally, steps that need to be taken include the
following:
1. Training. Personnel must be trained in their part of
the charting process. Calculating control limits, plotting
points and interpreting the charts may all be done by
different people, but they must understand their part.
2. Measurement. The measurement system must be in good
working order. The measuring devices must be appropriate
and accurate, the personnel making the measurement must be
19
trained properly. Evaluating the measurement system is
described in detail in the fourth chapter.
3. Setting up the control chart. Steps in setting up a
control chart are described in the next section.
Setting up a Control Chart
Specific steps must be taken in setting up any control
chart. Following these steps will increase the value of the
control chart.
1. Determine the characteristic to measure.
2. Choose the type of control chart.
3. Determine sample size and sampling frequency.
4. Collect data for 20 to 25 subgroups.
5. Calculate trial control limits and check for control.
6. Exclude subgroups with assignable causes and
recalculate control limits. If an assignable cause cannot
be found for a certain subgroup, do not exclude its data.
Repeat this process until all out of control subgroups with
assignable causes are not used.
7. Plot new data as it is generated and monitor control.
8. React to out of control conditions.
Reacting to Out of Control Conditions
Once an out of control condition is identified, steps
must be taken to identify and eliminate the cause. The true
power of SPC in improving processes is by eliminating
20
assignable causes. Eliminating these factors will decrease
the variation in the process.
Once causes have been eliminated from the process, new
control limits should be computed and drawn on the control
chart, as in step 5 above.
Shewhart Control Charts
The Shewhart control charts were developed by Walter
Shewhart in the 1920's, and published in his book Economic
Control of Quality of Manufactured Product (1931). They are
intended to show what causes are affecting a process and
what changes to the process effect its outcome.
Shewhart control charts common characteristics include
their parallel centerline, upper and lower control limits.
All Shewhart control charts are evaluated the same way,
whether they are variable or attribute charts.
Pattern Analysis
The basis for deciding if a control chart is exhibiting
out of control conditions is by examination of possible
patterns. With the exception of PRE-Control, the charts
have common tests for natural and unnatural patterns.
The characteristics of a natural pattern are that the
plotted points fluctuate in a random chance pattern. They
should follow no pattern or recognizable system. The
21
characteristics of a natural pattern can be summed up as
follows:
1) None of the points exceed the control limits.
2) A few of the points spread out and approach the
control limits.
3) Most of the points are on both sides of the
centerline.
4) There appears no pattern or system in the points.
An unnatural pattern is marked by points that fluctuate
widely, exceeding the control limits, or appearing in non
random patterns. These patterns are usually one of the
following:
1) A single point exceeding the three-sigma control
limit (points 10 and 21 in Figure 1).
7
8
9
10
11
12
13
UCL
CL
LCL
X
X
Figure 1. Out of Control Condition #1
22
2) Two out of three points exceeding two-sigma from the
centerline or four out of five exceeding one sigma.
6
7
8
9
10
11
12
131 3 5 7 9 11 13 15 17 19 21 23 25
UCL
+ 2 σ
+ 1 σ
CL
− 1 σ
− 2 σ
LCL
X X
X
Figure 2. Out of Control Condition #2
23
3) Eight successive points on one side of the
centerline.
7
8
9
10
11
12
13
UCL
CL
LCL
Figure 3. Out of Control Condition #3
24
4) Eight consecutive points within one standard
deviation of the center line.
6
7
8
9
10
11
12
13 UCL
+2 σ
+1 σ
CL
−1 σ
−2 σ
LCL
Figure 4. Out of Control Condition #4
25
5) A non random pattern - i.e. two successive points on
one side of the centerline, followed by one point on the
other, with the pattern of three repeated several times, or
any other pattern that corresponds to some non random
phenonoma. In Figure 5, the pattern is constantly up and
down. This might correspond to some manufacturing pattern,
perhaps a day and night shift. Patterns like this can be a
good clue to out of control conditions.
7
8
9
10
11
12
13
UCL
CL
LCL
Figure 5. Non-Random Pattern
26
Variables Control Charts
Introduction
Shewhart control charts designed for variables data are
the X-bar and R chart and the individuals chart. They are
the most powerful of the control charts and generally
require the smallest sample sizes.
X-bar and R Charts
X-bar and R measure two characteristics of the process
at the same time. The central tendency, or mean of the
process is measured on the X-bar chart, with the variability
or spread of the chart measured on the R chart. The symbols
X-bar and R stand for average and range, respectively. An
X-bar and R chart can be constructed for different levels of
sensitivity. We will concentrate on the standard level of
sensitivity, the three sigma control chart.
The basic procedure for constructing an X-bar and R
chart is to sample the process at preselected intervals,
compute the average and range of the sample, and plot the
two points on the chart. After enough samples have been
generated, control limits are computed and drawn on the
chart. All points are compared to the control limits to
detect any out of control conditions, which would signal
that something non-normal or out of the ordinary has
27
happened to the process. Any out of control conditions
should start investigation into causes, and elimination of
those causes if they are detrimental to the product.
Construction
The control lines for the 3σ X-bar chart are:
Centerline = X
RX A2UCL +=
LCL = −X RA2
The control lines for the R chart are:
Centerline = R
UCL = 4D R
LCL = 3D R
The symbols A2, D3 and D4 are statistical constants that
can be found in the tables in the appendix. Their values
vary depending upon the sample size. For example, with a
sample size of three, A2 would be 1.023; D3 would be zero
and D4 would equal 2.574.
28
Sample Sizes
The most important aspect of designing an X-bar and R
chart is the sampling plan. An improper sampling plan will
invalidate the chart, and may cause the chart to be
misleading. Points that are out of control on the chart may
actually not be and vice versa.
When designing a sampling plan one must consider the two
types of variation shown on the chart. The X-bar chart
shows long term variation - variation that happens between
subgroups. The R chart shows short term variation -
variation that occurs within the subgroups. This
distinction between short term and long term variation is
very important.
Since the X-bar chart uses the range to set its control
limits, essentially using short term variation to predict
long term variation, any improper sampling will effect both
the X-bar and R charts.
It is important to get the short term variation within
the subgroup. For instance, on a press we can expect that
five consecutive samples will have the same ink density
(unless the press is two-around, etc.). Therefore, drawing
our sample from five consecutive samples will not give us a
true value of short term variation. However, five samples
from the folder to evaluate fold skew could very well be a
representative sample of short term variation.
29
The best approach is to determine what factors influence
short term variation and try to capture them in a sample.
On a folder, the number of jaws would be a factor; if a
press is two around we should try to include both halves of
the plate or blanket in the sample.
Because web printing is a continuous process, time is a
factor in short term variation. Try to determine how much
of an influence time is and work with it. For example,
pulling one book a minute for five minutes to form a
subgroup of five would be a be better alternative than five
in a row. Each situation will be different with factors
such as the speed of the press and the type of work
influencing the sampling plan.
Potential Applications
There are many potential applications for the X-bar and
R chart in printing. Ink density, dot gain, and cutoff are
examples. There are however, some advanced charts that may
be more applicable in detecting the small shifts that occur
in the printing process.
One chart that may be more useful in detecting small
shifts in the process is the Cumulative Sum, or CUSUM chart.
This chart is constructed quite differently than an X-bar
and R chart, using a V-shaped mask instead of control lines.
It is quite often used in conjunction with an X-bar chart.
This control chart is quite advanced, and will not be
30
discussed in detail here. Several of the books in the
bibliography discuss CUSUM charts in detail.
Ink density may be measured throughout a run and
plotted on an X-bar and R chart. This would give us two
pieces of information; the average ink density and the
normal variation for the process. If, once color was set,
we plotted ink density on a control chart, and only made
adjustments when the chart indicated out of control
conditions, we would have used the variation of the process
to control the process. Out of control conditions would be
investigated - and hopefully eliminated from the process.
For example, the ink density dropped below the lower control
limit - what is the cause? Did we run out of ink, is there
water in the ink, etc. Once these conditions are
identified, we can work to eliminate them from reoccurring.
In the same way we can track dot gain through a run.
In this case an out of control condition might signal some
other cause - emulsified ink, piling, etc.
Cutoff, being a mechanical condition, is more easily
controlled by SPC. Many times operators make adjustment to
the cutoff control based on the condition of one book. Use
of a control chart would eliminate unnecessary moves, and
actually reduce the cutoff variation in the final product.
Example: Midtone dot gain.
A press operator wished to construct an X-bar and R
chart for midtone dot gain. Because he had a two around
31
press he chose a sample size of four and a sampling
frequency of fifteen minutes. The data for his first twenty
subgroups is summarized below.
Subgroup
1 2 3 4 5
1 23.32 24.74 21.64 21.19 23.15
2 24.41 23.48 24.67 22.78 24.88
3 22.01 19.64 22.81 22.16 22.89
4 19.16 24.11 19.74 22.33 22.56
X 22.22 22.99 22.22 22.11 23.37
R 5.25 5.10 4.93 1.59 2.32
6 7 8 9 10
1 24.08 23.03 25.81 22.70 20.44
2 21.20 24.70 18.67 24.61 21.80
3 18.58 18.57 20.17 20.58 23.07
4 24.30 19.56 19.14 19.91 22.52
X 22.04 21.46 20.95 21.95 21.96
R 5.72 6.13 7.13 4.70 2.63
32
11 12 13 14 15
1 21.92 22.33 23.36 21.38 23.20
2 21.27 21.87 21.71 23.49 22.38
3 22.55 24.03 21.23 22.82 23.32
4 23.42 23.76 20.59 20.89 21.71
X 22.29 23.00 21.72 22.14 22.65
R 2.15 2.15 2.77 2.60 1.61
16 17 18 19 20
1 21.53 23.01 22.15 22.26 20.56
2 21.23 20.06 22.86 20.36 23.69
3 22.70 22.67 23.81 23.82 20.53
4 22.41 24.03 21.07 20.95 22.56
X 21.97 22.44 22.47 21.85 21.83
R 1.48 3.97 2.74 3.46 3.16
Table 1. X-bar and R Chart Example Data
The operator looked up his constants for a subgroup of
size four and found A2 = 0.729, d3 = 0, and d4 = 2.282. He
then calculated his 3σ control limits and came up with the
following results:
33
X-bar chart R Chart
UCL = 24.8 UCL = 8.2
X = 22.2 R = 3.6
LCL = 19.6 LCL = 0
His finished control chart appears below:
Dot Gain X-bar Chart
18
20
22
24
26
28
CL
LCL
UCL
Dot Gain R Chart
0
2
4
6
8
10
CL
UCL
LCL
Figure 6. X-bar and R Chart Example
Since the charts did not exhibit any signs of out of
control conditions, the press operator did not make any
34
adjustmens. Any out of control conditions on the charts
would prompt the operator to determine and correct the
cause.
The X-bar and R chart can be a great tool to understand
and reduce the variation in a process.
Individuals Charts
Individuals charts are based on one measurement, rather
than a subgroup as with X-bar and R charts. This makes
these charts easier to construct, but less powerful.
Sometimes it may be necessary to use an individuals chart,
such as when it is expensive or time-consuming to collect
more than one measurement. Examples might include ink
mileage or afterburner efficiency.
Construction
The control lines for a 3σ individuals chart are:
Centerline X=
UCL X 2.88R= +
LCL X 2.88R= −
35
Potential Applications
Potential applications include any data that is hard to
gather, or takes too long to gather. Examples could include
ink mileage, paper waste, etc.
An ink manufacturer wanted to track the mileage on his
black ink. He decided to use one months data to compute
pounds of ink per thousand copies. His data looked like
this:
Jan Feb Mar Apr May
1.22 1.22 1.08 1.07 1.15
June July Aug Sept Oct
1.19 1.14 1.26 1.13 1.03
Table 2. Individuals Chart Example Data
He computed his X to be 1.15 and his R to be 0.07. His
control limits turned out to be:
UCL = 1.37
LCL = 0.93
His finished control chart follows:
36
Ink Mileage Individuals
0.800.901.001.101.201.301.401.50
UCL
LCL
CL
Figure 7. Individuals Chart Example
For applications where an X-bar and R chart is not
feasible, an individuals chart can be used to get a handle
on variation.
Attributes Control Charts
Introduction
Attributes control charts are based on data that is
countable rather than measured. Counts may include plate
scratches, number of books with skewed fold or number of
short skids.
Attribute control charts are further divided into two
types - those based nonconforming units, and those based on
nonconformities. This difference is important, and can be
quite confusing.
37
A nonconforming piece, or defective part, is a single
piece that is not good. It may contain one or more defects
or nonconformities that make it so. We can count the
defective part or we can count the number of defects on it.
An example would be a plate with holes. We can count
the defective plate (a nonconforming item) or we can count
the number of holes on the plate (nonconformities). Each
count has its own type of control chart.
Sampling Plans
Because attribute charts are not based on measurements,
the sample sizes need to be larger to get the same
sensitivity. Sampling plans should be put together to
capture the normal short term variation in the subgroup.
Typical sampling plans for attribute charts include part
related plans such as batches, and time related plans such
as days or shifts.
Fraction Nonconforming - p-, np- Charts
P-and np- charts are based on number of defectives.
They may be used for monitoring percent defective. P-charts
are based on percent defective, if the sample size is
consistent np-charts based on number defective may be used.
38
p-Charts
P-charts are so named because they track percentages.
The p-chart uses a percent defective as its values.
Construction
The control lines for the 3σ p-chart are:
Centerline p=
( )/np1p3pUCL −+=
( )/np1p3pLCL −−=
Potential Applications
For example, a plateroom wished to track defective
plates. The manager decided on a p-chart. His subgroup
size would be however many plates were made each day. He
would count the number of bad plates and find the
percentage. His data is below.
39
1 2 3 4 5
d 11 9 14 11 10
n 150 150 150 150 150
p 0.073 0.060 0.093 0.073 0.067
6 7 8 9 10
d 14 10 9 12 10
n 150 150 150 150 150
p 0.093 0.067 0.060 0.080 0.067
Table 3. p-chart data
The control limits were calculated:
p = 0.0733
UCL = 0.137
LCL = .009
40
His finished control chart looked like this:
Plate Defect p-chart
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1 2 3 4 5 6 7 8 9 10
CL
UCL
LCL
Figure 8. Plate Defect p-chart
np-Charts
The np-chart is just like the p-chart except it uses the
number of defective items rather than the percentage. This
is only possible if each sample has the same number of
units.
Construction
The control lines for the 3σ np-chart are:
Centerline np=
( )p1pn3pnUCL −+=
41
( )p1pn3pnLCL −−=
Potential Applications
The potential applications are the same as for p-charts.
The same data for the p-chart could be constructed as a
np-chart since the sample sizes were the same. If the
percent defective remained the same each day, the control
lines would be:
np = 11
UCL = 20.57
LCL = 1.43
The control chart would now look like this:
42
Plate Defect np-chart
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
CL
UCL
LCL
Figure 9. Plate Defect np-chart
Nonconformities - u-, c- Charts
The charts based on number of defects are the u- and c-
charts. They are used when the number of defective units is
not as important as the number of defects.
u-Charts
U-charts are most useful when several types of defects
can occur in one unit. The u-chart measures defects per
unit.
43
Construction
The control lines for the 3σ u-chart are:
Centerline u=
UCL u 3 u / n= +
LCL u 3 u / n= −
Potential Applications
Since holes were a major cause of plate defects, the
plateroom manager decided to track the number of holes in a
sample of plates. For this, he chose a u-chart. He
randomly selected ten plates from ten consecutive shifts to
start his control chart. His data looked like this:
44
1 2 3 4 5
Holes 12 14 12 12 12
Plates 10 10 10 10 10
Holes/plate 1.2 1.4 1.2 1.2 1.2
6 7 8 9 10
Holes 11 13 13 14 17
Plates 10 10 10 10 10
Holes/plate 1.1 1.3 1.3 1.4 1.7
Table 4. Holes/Plate Example Data
He calculated his control lines to be:
u = 1.3
UCL = 2.38
LCL = 0.22.
His u-chart looked like this:
45
Holes/Plate u-chart
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8 9 10
CL
UCL
LCL
Figure 10. Holes/Plate Example u-chart
c-Charts
The c-chart is a chart of counts. It differs from the
u-chart in that the defects are not counted by unit. An
example would be overall defects in a plate, rather than
holes per plate.
Construction
The control lines for the 3σ c-chart are:
cCenterline =
c3cUCL +=
c3cLCL +=
46
Potential Applications
The c-chart relates to the u-chart just as the np- and
p-charts relate. The c-chart is good when the sample size
is constant.
Advanced Control Charts
The advanced control charts we will discuss here are
short run SPC and PRE-Control. Short run SPC is a technique
that can be used when runs are not long enough to use a
conventional control chart. Different types of work, or
short runs may be kept on the same chart. PRE-Control is a
technique that can be used to monitor variables or output to
determine if they are within specification. It can be used
as a monitoring tool by the operator.
Short-Run SPC
Short run SPC can be a very powerful form of SPC in the
printing industry. Since runs are generally not conducted
over a period of months or even weeks, a short run chart can
be used where a conventional control charts could not. Data
to be used in a short run SPC chart is normalized from
existing data before being used. Depending upon the type of
process being charted, different data normalization
techniques should be used.
47
Data Normalization
The three types of data normalization are nominal,
target, and short run. The first two only take into account
deviation from nominal or specification, the second both
deviation and variation.
For processes using data from similar setups, such as
same paper stock and press, either the nominal or target
normalization may be used.
For processes where variation is very different from
setup to setup, the short run normalization is preferred.
This normalization takes into account the differences in
variability between setups.
Nominal
The data may be normalized by measuring the deviation
from the nominal of the specification. The range chart will
be unaffected but the centerline of the X-bar chart will be
very close to zero.
48
Target
The data in this case are coded by measuring the
deviation from an historical process average. The
centerline of the X-bar chart will again be very close to
zero.
Short Run
This technique takes into account the differences in
variability between different parts by dividing by the
historical average range for each part.
The normalization formulae are:
R HistoricalRR Coded =
R HistoricalX HistoricalXX Coded −
=
The control limits for the coded data, for a three sigma
control chart, appear below:
R Chart X-bar Chart
Centerline 1= Centerline 0=
UCL 4D= UCL 2A= +
LCL 3D= LCL 2A= −
49
X-bar and R Charts
Once data are coded, they may be plotted and interpreted
just like any other X-bar and R chart.
For example, if the data from the X-bar and R chart
example above are used, we can construct a nominal short run
chart. The data would be normalized to look like this:
Subgroup
1 2 3 4 5
1 3.32 4.74 1.64 1.19 3.15
2 4.41 3.48 4.67 2.78 4.88
3 2.01 -0.36 2.81 2.16 2.89
4 -0.84 4.11 -0.26 2.33 2.56
X 2.22 2.99 2.22 2.11 3.37
R 5.25 5.10 4.93 1.59 2.32
6 7 8 9 10
1 4.08 3.03 5.81 2.70 0.44
2 1.20 4.70 -1.33 4.61 1.80
3 -1.42 -1.43 0.17 0.58 3.07
4 4.30 -0.44 -0.86 -0.09 2.52
X 2.04 1.46 0.95 1.95 1.96
R 5.72 6.13 7.13 4.70 2.63
50
11 12 13 14 15
1 1.92 2.33 3.36 1.38 3.20
2 1.27 1.87 1.71 3.49 2.38
3 2.55 4.03 1.23 2.82 3.32
4 3.42 3.76 0.59 0.89 1.71
X 2.29 3.00 1.72 2.14 2.65
R 2.15 2.15 2.77 2.60 1.61
16 17 18 19 20
1 1.53 3.01 2.15 2.26 0.56
2 1.23 0.06 2.86 0.36 3.69
3 2.70 2.67 3.81 3.82 0.53
4 2.41 4.03 1.07 0.95 2.56
X 1.97 2.44 2.47 1.85 1.83
R 1.48 3.97 2.74 3.46 3.16
Table 5. Nominal Short Run Example Data
The new control limits are calculated to be:
X = 2.18
UCL = 4.79
LCL = -0.43.
51
The control chart looks like this:
Dot Gain Short Run X-bar Chart
-1.000.001.002.003.004.005.006.00
1 2 3 4 5 6 7 8 91011121314151617181920
UCL
LCL
CL
Figure 11. Nominal Example X-bar chart
The R-chart is unaffected by the normalization and would
look the same as the previous R-chart.
The data could also be normalized with both the
historical average and range. This would account for both
centering and variability. The data would look like this:
52
1 2 3 4 5
1 23.32 24.74 21.64 21.19 23.15
2 24.41 23.48 24.67 22.78 24.88
3 22.01 19.64 22.81 22.16 22.89
4 19.16 24.11 19.74 22.33 22.56
X 22.22 22.99 22.22 22.11 23.37
R 5.25 5.10 4.93 1.59 2.32
6 7 8 9 10
1 24.08 23.03 25.81 22.70 20.44
2 21.20 24.70 18.67 24.61 21.80
3 18.58 18.57 20.17 20.58 23.07
4 24.30 19.56 19.14 19.91 22.52
X 22.04 21.46 20.95 21.95 21.96
R 5.72 6.13 7.13 4.70 2.63
53
11 12 13 14 15
1 21.92 22.33 23.36 21.38 23.20
2 21.27 21.87 21.71 23.49 22.38
3 22.55 24.03 21.23 22.82 23.32
4 23.42 23.76 20.59 20.89 21.71
X 22.29 23.00 21.72 22.14 22.65
R 2.15 2.15 2.77 2.60 1.61
16 17 18 19 20
1 21.53 23.01 22.15 22.26 20.56
2 21.23 20.06 22.86 20.36 23.69
3 22.70 22.67 23.81 23.82 20.53
4 22.41 24.03 21.07 20.95 22.56
X 21.97 22.44 22.47 21.85 21.83
R 1.48 3.97 2.74 3.46 3.16
Table 6. Short Run Example Data
The new control lines are calculated to be:
X-Chart R-Chart
X = 0.44 R = 0.72
UCL = 0.96 UCL = 1.63
LCL = -0.09 LCL = 0
54
The charts appear as below:
D ot G a in S hort R un X -bar Chart
-0.2
0
0.2
0.4
0.6
0.8
1 UCL
LCL
CL
D ot G a in S hort R un R Chart
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
CL
UCL
Figure 12. Short Run X-bar and R Chart
55
PRE-Control
Introduction
Once control of a process is established, and
specifications are set it may be necessary to transfer
control charts to line personnel. In some cases control
charting may be too complicated, time consuming or
unnecessary. In these cases, a simple technique call PRE-
Control may be used.
PRE-Control compares parts not to statistically
calculated limits but to the specifications. PRE-Control is
not meant for establishing statistical control, but for
maintaining the process between a set of specifications.
Construction
A PRE-Control chart appears on the following page:
56
PRE-Control
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10 11
USL
LSL
Red Zone
Red Zone
Yellow Zone
Yellow Zone
Green Zone
Figure 13. PRE-Control Chart
The red bands in the chart represent out of
specification measurements. The process must be adjusted
for no more out of specification product to be produced.
The yellow bands are cautionary zones where the process may
have to be adjusted. The green band is a zone of good
product.
The line marking the boundary between the green and
yellow zones is one half the distance between the nominal of
the specification and the tolerance limit. The line marking
the boundary between the yellow and red zones is the
specification limit. With these lines the chart will have a
green zone equal to one-half the specification, two yellow
zones each equal to one-quarter the specification, and two
red zones.
57
Use of PRE-Control
There are only three steps that need be done in using a
PRE-Control chart:
Qualify the setup. Every piece must be measured until
five greens in a row are produced. If one yellow or red is
encountered, restart the count. Make any adjustments
necessary during this period to produce five greens in a
row.
Run. Once the process is qualified, sample and measure
two consecutive pieces periodically. Plot both pieces on
the PRE-Control chart in the appropriate band. The
measurements do not need to be precisely plotted, they just
need to be in the appropriate band. If one of the following
conditions occur the process must be stopped or adjusted:
One red: The process is already producing bad product
and must be stopped and corrected.
Two yellows: The process should be adjusted back to the
center. If the yellows are in opposite zones a more
sophisticated investigation might have to be undertaken.
If the process is adjusted, it will need to be
requalified. That is five greens in a row will have to be
produced to ensure the setup is correct.
A big advantage of PRE-Control is that it is much easier
to use than control charts. If gages are made that have the
58
appropriate colors on them the operators need not even worry
about precise measurements.
Potential Applications
PRE-Control is applicable whenever product can be
sampled and specifications can be set up. It is important
to remember it is not a tool to monitor statistical control,
but to keep product within the specifications. Applications
could include fold, plate burning, etc.
An example, the plateroom manager wished to monitor the
consistently of his light sources. Since he had
specifications set, he decided to use a PRE-Control chart.
His specifications were 30 - 35 on a continuous scale. He
first burned several plates until he got five greens in a
row. He then started his PRE-Control chart.
His PRE-Control chart was set up as follows:
Red Zone - > 35
Yellow Zone - 33.75 - 35
Green Zone - 31.25 - 33.75
Yellow Zone - 30 - 31.25
Red Zone - < 30.
At the beginning of each shift, a platemaker would burn
a scale on two consecutive plates. The chart for ten shifts
was as follows:
59
The chart looked like this:
Plate Exposure PRE-Control
27
29
31
33
35
37
1 2 3 4 5 6 7 8 9 10 11
USL
Green Zone
LSL
Red Zone
Red Zone
Yellow Zone
Yellow Zone
Figure 14. Example PRE-Control Chart
The first points out violating the PRE-Control rules
occur at shift ten. Both points are below the red line, so
the exposure unit should be adjusted back toward the center.
Once this is done, the setup will need to be qualified
again.
Conclusion
Statistical Process Control can be a major factor in
process improvement. It can be used to monitor any process
characteristic, input or output. A good SPC program can put
processes in control, which is generally great improvement
60
by itself, and from there other methods of improvement, such
as Design of Experiments, explained in the next chapter, can
improve it even more.
61
CHAPTER 3: DESIGN OF EXPERIMENTS
Introduction
Design of Experiments is a name for a set of methods
that aid in determining the outputs that occur for a given
set of inputs. The inputs are purposely varied to determine
their individual and collective action on the output.
Terminology will be discussed, along with the different
types of experiments, how to design and apply them, and how
to interpret the resulting data. Taguchi methods will be
discussed as the major tool for use in experimental design.
Fundamentals of DOE
Terminology
Designed Experiment. An experiment where variables are
manipulated according to a predetermined plan, and the
resulting data are analyzed statistically to determine the
effects of any variable or combination of variables.
Response Variable. The output variable, or the variable
being investigated, also called the dependent variable.
62
Factors. The input variables or the variables that are
intentionally varied, also called the primary or independent
variables.
Random Variables. Variables which, although identified,
either cannot or should not be deliberately held constant.
Experimental Error. The variables that are not
identified or controlled. They are analogous to the "common
cause" variables of SPC. Because the term "error" has a
negative connotation, the term "all others" will be used
here, since this is really the contribution of all factors
not controlled. Sometimes these factors are called "noise"
factors, since they obscure the true effects of the factors.
Replication. Repeating a set of conditions in an
experiment.
Interaction. Condition in which the effect of one
factor depends upon the level of another factor.
Level. The values of a factor being studied in the
experiment.
Steps in Design of Experiments
In order to be successful, the designed experiment must
follow some logical sequence and meet some specific
criteria. A poorly planned experiment, no matter how well
it is carried out, will not have the statistical validity
necessary to come to a conclusion. The following are steps
necessary in using a designed experiment.
63
1. Clearly identify the problem. The problem (what we
wish to measure) must be clearly identified. It also must
be specific, e.g. "Too much dot gain on Press 1" would be a
specific problem.
2. Determine the response variable and how to measure
it. The response variable is the variable that will be
measured to determine the effects of the factors. It must
be measurable. Stating the response variable and how it is
measured is a good idea, e.g. "The response variable will be
dot gain, measured by densitometer."
3. Identify factors of interest and possible
interactions. This step determines the variable that will
be used in the experiment. It is sometimes a good idea to
use a group brainstorming session to come up with
appropriate factors. Factors should be weighed against each
other for possible interaction before picking the
experimental design. The number of factors chosen will
determine the cost of the experiment.
4. Select representative levels of each factor. Levels
for each factor should be chosen that a representative of
normal conditions. For example, if blanket packing is a
factor, representative levels might be .008" and .012", not
.05". In the Taguchi methods, two level experiments are
generally favored.
64
5. Pick an appropriate experimental design. The
experimental design chosen is determined by the number of
factors and interactions chosen in step 3.
6. Run the test and gather the data. The test must be
ran according to the design, and the response variable
measured.
7. Graph and interpret the results (graphical analysis).
Graphical analysis is relatively easy, and may show
responses and interactions harder to see in the statistical
analysis.
8. Determine confidence and each factors contribution
(statistical analysis). Graphical analysis does not
calculate the contribution of each factor or interaction,
and will not show the confidence in the results. Generally,
statistical analysis is done using Analysis of Variance.
9. Run addition tests for confirmation or refinement, as
necessary. Once results have been formulated, a
confirmation test should be run to eliminate the possibility
of a wrong conclusion.
10. Implement the improvements. Once confirmed, the
results must be implemented on an ongoing basis.
65
Statistical Analysis
One Way ANOVA
Analysis of Variance (ANOVA) is the statistical tool
used to determine the probability that values from two
samples come from different populations. In other words, it
measures the probability that levels of a certain factor
actually create different results.
An ANOVA compares the variation between levels of a
factor against variation within those levels. The greater
the ratio of "between level" versus "within level" the
higher the probability there is actually a difference
between the levels.
Results from an ANOVA are summarized in an "ANOVA table"
which is shown below. The terms are explained below the
table.
Source of
Variation
Degrees of
Freedom1
Variation
(SS)2
Variance
(MS)3
F4 α5
Between Level
Within Level
Total
Table 7. One-Way ANOVA Table
66
1. The Degrees of Freedom depicts the assurance the
variance is close to the true population variance It is
generally one less than the number of values used to compute
the sum of squares.
2. The variation is the Sum of Squares.
3. The variance is the Mean Square - the Sum of Squares
divided by the Degrees of Freedom.
4. F is the ratio of the between level variance divided
by the total variance. This is used as a lookup to find the
probability.
5. Alpha is the probability that there is a difference
between the levels.
Two Way ANOVA
The two way ANOVA is similar to a one way ANOVA, but
compares the effects of more than one variable. For
instance, if an experiment had factors of plate and fountain
solution, a two way ANOVA would be appropriate. The table
is similar, but has rows for each variable, plus one for all
other variables.
67
Source of
Variation
Degrees of
Freedom
Variation
(SS)
Variance
(MS)
F α
Factor One
Factor Two
...
Factor n
All Others
Total
Table 8. Two-Way ANOVA Table
Analysis of Variance is a complicated subject and will
not be covered here. Several good textbooks are listed in
the bibliography. A table for finding α is included in the
appendix.
Graphical Analysis
Results from experiments may be analyzed graphically by
plotting them on main effect plots and interaction plots.
As their names suggest, these plots analyze the effects of
the main factors and the first level interactions.
Graphical plots may make it easier to visualize the effects
of the factors and may lead to quick conclusions about the
experiment.
68
Graphical analysis is not a substitute for statistical
analysis, but a supplement to it. Both graphical and
statistical analysis should be performed on each experiment.
Main Effect Plots
Main Effect Plot
05
101520253035
. A1 A2 A3
Figure 15. Main Effect Plot
The main effect plot shows the responses of each of the
main variables. It is simply plotted like a horizontal
histogram for each factor level.
69
Interaction Plots
Interaction Plot
0
2
4
6
8
10
12
B1 B2
A1
A2
Figure 16. Interaction Plot
In Figure 2, the interaction plot shows an interaction.
That is, the effect of A changes depending upon the level of
B. This is demonstrated by the crossed lines. The more
nearly perpendicular the lines, the greater the interaction.
Interaction Plot
02468
10121416
B1 B2
A1
A2
Figure 17. Interaction Plot
70
In Figure 3, the interaction plot shows an absence of an
interaction. This is demonstrated by the parallel lines,
indicating the effects of A and B do not change because of
the others level.
One Factor Experiments
The most common type of experiments are one factor
experiments, in which only one variable is of interest. An
example would be a plate test, in which one variable (the
type of plate) is varied. This type of experiment would use
a one-way ANOVA.
If a printer wanted to test two plates for midtone dot
gain, a one factor experiment would be in order. The data
from such a test could look like this:
71
Run Plate 1 Plate 2 Run Plate 1 Plate 2
1 34.59 30.72 13 33.74 30.71
2 34.50 29.86 14 32.66 29.66
3 32.81 28.99 15 33.66 29.86
4 32.56 29.31 16 32.74 30.95
5 33.63 29.68 17 33.55 29.82
6 34.14 28.85 18 33.08 28.51
7 33.56 32.16 19 33.14 29.26
8 35.08 31.65 20 32.21 30.92
9 33.39 29.43 21 34.31 29.59
10 32.15 29.98 22 33.81 30.88
11 32.96 27.98 23 34.46 28.58
12 33.18 28.87 24 33.43 29.96
25 34.57 29.94
Table 9. One Factor Experiment Example Data
72
A one-way ANOVA should be performed on the data. The
ANOVA would look like this:
Level Count Sum Average Variance
Plate 1 25 837.90 33.52 0.618643
Plate 2 25 746.12 29.84 1.002565
Source SS df MS F Alpha
Between
Groups
168.46 1 168.46 207.82 0
Within Groups 38.909 48 0.810
Total 207.37 49
Table 10. One Factor Example ANOVA Table
The low Alpha (zero) shows that there is a big
difference between the two plates.
Factorial Experiments
A factorial experiment is one where more than one factor
is under control, and all are varied at the same time. A
full factorial is where at least one result is taken from
each combination of levels that can be formed from the
different factors.
73
These experiments are the most revealing of any designs,
but with this comes the penalty of complexity, and cost.
Taguchi designs, discussed later, offer many of the
advantages, at a cheaper cost, than full factorial designs.
Full factorial experiments could be used when several
factors, each at several levels, are to be tested to find
the optimum settings.
The Taguchi Methods
Introduction
Perhaps the most effective of all the DOE tools is a
relative newcomer, the Taguchi methods. It is increasingly
being accepted by quality professionals across the United
States, and is in widespread use in Japan.
It is not the most statistically correct of methods, and
many statisticians invalidate it for that reason, but it is
a very effective tool, and translates well to use in all
industries. It is for this reason that this paper will
emphasize its use above all others.
The concepts, tables and terms can be daunting, but they
have been simplified as much as possible, with several
tables not found anywhere else included.
74
Dr. Genichi Taguchi is a Japanese statistician and
engineer. He studied several classical methods of
experimentation, and rejected them as not appropriate for
use in industrial situations, and then developed his own
methods. He was awarded the Deming Prize in 1960 for his
contributions to the field.
The Taguchi Methods are divided into three parts. The
first is his concepts of the Loss Function. Its relevance
is not as an experimental tool, but as a concept that any
deviation from a nominal target has a monetary loss
associated with it, even if it within specifications.
The second of the concepts is the most important, and is
Taguchi's concept of the orthogonal array, and the linear
graph. These are the two tools used extensively in the
Taguchi methods. The orthogonal array is a matrix that
shows which combinations of levels should be used in each
experimental run. A linear graph shows which columns in the
array show interactions between variables.
The next concepts in the Taguchi philosophy are
parameter and tolerance design, and the use of Signal-to-
noise (S/N) ratios. These tools can be valuable in the
later stages of quality improvement, and will be discussed
later in this chapter.
75
Limitations
There are several limitations to the Taguchi design.
One of them is strongly criticized by classical
statisticians, and that is the element of randomization.
Randomization of experimental run is way to try and even out
the effects of non-controlled variables. Not randomizing
runs of an experiment may cause confounding because a non-
controlled variable may change levels in the same pattern,
and may disrupt test results.
The next limitation to the Taguchi design has been
turned into an advantage. Taguchi assumes that second order
interactions are not significant. That is, interactions
between three or more variables do not occur. This can be a
limitation if second order interactions are present, but the
design makes this assumption into an advantage. By assuming
no second order interactions exist, we can use many fewer
runs to test the main effects and first order interactions.
Orthogonal Arrays
The orthogonal array is a matrix used to design and
analyze Taguchi experiments. They are normally designated
by the letter "L" and a number indicating the number of runs
necessary to complete the experiment. For example, an L4
(the simplest of the arrays) has four runs, and looks like
this:
76
Variables
Run Number 1 2 3
1 1 1 1
2 1 2 2
3 2 1 1
4 2 2 1
Table 11. L4 Array
The three columns are reserved for each variable, up to
three in this case. The numbers 1 and 2 in each column show
which levels are supposed to be used for each of the four
runs.
Linear Graphs
The linear graph is a method to show which columns show
the interactions between variables. These graphs can get
quite cluttered, and very confusing, so we will use another
table to show the interactions for each array. An example
for the L4 array is shown below:
Variables
Interactions 1 2 3
2x3 1x3 1x2
Table 12. Interaction Table
77
This interaction table shows that the interaction
between columns 1 and 2 (and therefor any variables assigned
to them) is contained in column three).
These tables will be combined into one table, which will
show interactions and runs. It will have room for variable
and level assignments, and can be used in planning, running
and analyzing the experiment. These tables are very helpful
compared to the simple array and linear graphs.
Assigning Factors
Perhaps the most important part of a Taguchi experiment
is assigning the factors and interactions to the appropriate
columns. An error in design of the experiment can cause
confounding and make the experiment invalid.
A list of factors and interactions should be used when
assigning columns. The factors should be checked off and
they are assigned and checked for unintended interactions.
A way to make this easier is to arrange the list with
factors that have interactions at the top of the list - they
will be assigned first.
For example, we are going to use an L8 with four factors
and one interaction. Our factor list is as follows:
Fountain Solution
Blanket Height
Oven Temperature
Plate Packing
78
Interaction of Blanket Height and Plate Packing
It will be advantageous to rearrange this list with
factors that have interactions at the top, like this:
A = Blanket Height
B = Plate Packing
C = Fountain Solution
D = Oven Temperature
AB = Interaction of Blanket Height and Plate Packing
With this order we can assign our columns as follows:
Place Factor A in column 1.
Place Factor B in column 2
Since column 3 is the interaction of columns 1 and 2, it
is the interaction of Factors A and B (Blanket Height and
Plate Packing), and therefore our AB interaction should go
in column 3.
Factors C and D can go in any remaining column, since
any interactions also in those columns we do not expect to
happen. For example, if we place Fountain Solution in
column 4 and Oven Temperature in column 5, column 4 also
includes the interaction between columns 1 and 5 (Blanket
Height and Oven Temperature). Since this is not an
interaction we expect to happen, we can let this go.
79
Analysis
Once the experiment has been completed there should be
one result for each run in the array. These results need to
be analyzed statistically. A two-way ANOVA is the method
used.
First, the sum of squares for each column (factor) needs
to be found. The easiest way is to find the average
response for each level of the column. That is, find the
average response for level 1 of column A and level 2 of
column A. The square of the difference between these
results can be multiplied by half the number of runs to
determine the sum of squares. For example,
Number of Runs Multiply by
4 2
8 4
12 6
16 8
32 16.
The equation for this would be:
( ) runs ofnumber thehalf one *21=SS2
LL −
A table should be constructed for the sum of squares.
Using our previous example, the table would be:
80
Column Source Level 1 Level 2 Diff SS
1 A 29.95 30.79 0.84 1.411
2 B 30.26 30.48 0.22 0.097
3 AB 26.17 34.57 8.4 141.1
4 C 30.21 30.53 0.32 0.205
5 D 30.7 30.04 -0.66 0.871
6 All Others 29.42 31.32 1.9 7.22
7 All Others 30.77 29.97 -0.8 1.28
Table 13. Sum of Squares example table
This sum of squares value would then be used in the two-
way ANOVA. All the All Others columns (in this case columns
6 and 7) would be added together to produce the All Others
sum of squares. Their degrees of freedom would then be
however many columns they were. In this example, since
there are two columns of All Others, its degrees of freedom
would be 2.
81
The ANOVA table for this example would be:
ANOVA
Source SS df MS F Alpha
A 1.41 1 1.41 0.33
B 0.10 1 0.10 0.02
AB 141.12 1 141.12 33.20 0.031
C 0.20 1 0.20 0.05
D 0.87 1 0.87 0.20
All Others 8.50 2 4.25
Total 152.20 7
Table 14. L8 ANOVA Table example
The Alpha value is looked up in a table of F values. 1
- Alpha is the chance that source has a significant impact
on the response. In this case the AB interaction has a 97%
chance of being significant to the response. Controlling
the AB interaction (and therefor controlling A and B) will
give the response desired. The levels of A and B can be set
to those in the experiments which gave the desired response.
In this case, to minimize dot gain, A and B would both be
set to level 1.
L12 No Interactions
A special type of Taguchi design is the L12 array. This
array has no interactions, so it can be used for up to
82
eleven variables. It is a good design for screening
variables to determine their significance. Significant
variables can then be used in another experiment.
The L12 array is used just like any other Taguchi array,
including the L8 above. It is simpler to set up and
analyze, due to having no interactions.
S/N Ratios
Perhaps the most significant contribution of the Taguchi
methods is that of the Signal-to-Noise (S/N) Ratio. This is
a method of combining both mean output and variation into
one statistic. This is done by means of the following
formulae. There are two formulae to be used - smaller is
better and larger is better. In the case of trying to hit a
certain target, the deviation from that target should be
used as the input to the smaller is better formula.
Smaller is better -
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∑−= 2110log10/ iyn
NS
Larger is better -
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−= ∑
=
n
i iynNS
1
2
1011log10/
The idea behind the signal-to-noise ratio is that the
factors may be divided into three groups - (1) those that
affect both the mean level and the variability of the
83
response, (2) those that affect the mean level of the
response, but do not have much impact on the variability of
the response and (3) those factors that do not affect either
the mean level or the variability of the response. Factors
of the first type may be used to minimize the variability of
the process, factors of the second type may be used to set
the process mean, and factors of the third type may be
ignored.
Parameter Design
Parameter design takes the idea of Signal-to-Noise
ratios one step farther. It will analyze responses to
deliberately planned "noise" inputs in order to determine
the process variable values that will keep the process on
target and do so despite the "noise."
This process of parameter design gives a process that
will have a certain response no matter what "noise" it
actually encounters. A process in this state is said to be
"robust." Since the noise is not under control of the
process, being robust to it is an excellent characteristic.
A Taguchi design may be modified for parameter design.
Two arrays will be combined to form a matrix with the
controlled factors on the left and the noise factors on the
right. The factors under control are the "inner array",
while the noise factors are the "outer array."
84
The following example will use an L8 for the factors
under control and an L4 for the noise factors. The factors
identified as controllable are:
A = Chill Speed
B = Nip Size
C = Press Speed
D = Infeed Tension
The factors determined not to be under control are:
F = Paper Basis Weight
G = Paper Caliper
The following matrix is developed using the L8 of the
first factors and the L4 of the second. This will determine
the effects of A through D on fold cutoff. For each run of
the L8, four tests are to be made, one for each condition of
the L4.
85
F 1 2x3 1 1 2 2
G 2 1x3 1 2 1 2
Factor 3 1x2 1 2 2 1
Assign A B C D
Factors 1 2 3 4 5 6 7
2x3 1x3 1x2 1x5 1x4 1x7 1x6
Interacts 4x5 4x6 4x7 2x6 2x7 2x4 2x5
6x7 5x7 5x6 3x7 3x6 3x5 3x4
1 1 1 1 1 1 1 1
2 1 1 1 2 2 2 2
3 1 2 2 1 1 2 2
4 1 2 2 2 2 1 1
5 2 1 2 1 2 1 2
6 2 1 2 2 1 2 1
7 2 2 1 1 2 2 1
8 2 2 1 2 1 1 2
Table 15. Parameter Design Arrays
There will be 32 test results obtained from this
experiment. They will be analyzed using the S/N ratios (in
this case for smaller is better). The resultant S/N ratios
will be used in the ANOVA instead of the experimental
values.
86
The results obtained were:
Run S/N
1 53 50 53 50 34.26
2 27 31 27 31 29.19
3 56 52 56 52 34.63
4 35 30 35 30 30.16
5 28 26 28 26 28.61
6 47 58 47 58 34.26
7 56 58 56 58 35.11
8 58 51 58 51 34.67
Table 16. Parameter Design Example Data
We should now perform an ANOVA with the S/N ratios
calculated above. Doing this would show us that factor D
has a 1 - Alpha of 91% and factor B of 61%. Controlling
these two factors will give us the best chance of
controlling the output of the process - no matter what the
noise levels are.
This process will give us the set of factors that are
the most robust - the ones that keep us close to zero cutoff
deviation despite basis weight and caliper.
87
Conclusion
Design of experiments is a very powerful tool that has
not been used extensively in the printing industry. The
reduction of variation has a great impact on quality and
productivity, and design of experiments is a great tool in
reducing variation.
88
CHAPTER 4: PROCESS CAPABILITY AND MEASUREMENT
Introduction
Statistical Process Control and Design of Experiments
are good tools to tell us how our process is doing and what
we can do to minimize its variation; however, without good
measurements both techniques are useless. Measurement
science can help minimize the variation in the measurements
system, both in the gage and the inspectors. Several easy
to use techniques will be explained to help in this process.
Process capability is a buzzword in the quality industry
right now. It encompasses a wide range of subjects, from
ratios such as Cpk to studies made to minimize variation.
Process capability explains the process in terms of the
specifications.
Process Capability
Process capability refers to the normal behavior of a
process when operating in statistical control. It is
important that the process be in statistical control when
referring to process capability. Process capability is the
best the process could operate in without intervention from
external sources of variation. The actual performance of a
process will probably include external sources of variation
89
- eliminating this variation will give the true process
capability. Determining unusual sources of variation is
done with the use of control charts.
Definitions
Process Capability. The ability of a process to
consistently achieve desired results. The number associated
with process capability is 6σ. Process capability can only
be calculated for a process in statistical control.
Process Capability Ratios. Ratios, such as Cp and Cpk
compare the process with the specifications.
Tolerance. Tolerance is also called the specifications.
Process Capability Using a Histogram
Perhaps the easiest way to determine process capability
is by use of the histogram. A histogram can be plotted and
compared to the specifications to see what portion, if any,
of the histogram falls out of specification.
The normal technique for this is to take a sample of
about fifty units, during which time no adjustments to the
process are made and plotting a histogram. Upper and lower
tolerance limits should be drawn on the histogram. These
can be compared with the distribution to see if any fall out
of specification.
The histogram should be examined for the following
traits:
90
Centering. This defines the aim of the process.
Width. This defines the variability about the aim.
Shape. The shape of the histogram can reveal much about
the process. Histograms with two peaks show that two
populations have been mixed, other patterns reveal other
traits.
Common histogram shapes include the following:
Symmetrical. A symmetrical distribution is
characterized by the general shape of a normal distribution.
Skewed. Skewed distributions have more points on one
side of the center than the other.
Bimodal. Bimodal distributions have two peaks. This is
commonly caused by mixing two distributions.
Truncated. Truncated distributions are characterized by
their sudden drop-off in frequency. Many time this occurs
at the specification, indicating that true values are not
being recorded.
A chronological plot (a run chart or control chart) may
also be used to discover reasons for variability, such as
drift, cyclical changes or inconsistency.
Estimating Natural Process Limits
Natural process limits, limits that show where the
process can reasonably be expected to vary within, can be
constructed from data from the process. The natural process
91
limits are sometimes referred to as process capability and
the formula is
Process Capability 6= σ
where σ is the standard deviation from a process under
statistical control.
Processes under statistical control can compare the
natural tolerance directly to the specifications. This
underlies the importance of a process being in statistical
control before judgments of capability are made.
Process Capability Ratios
Process capability ratios compare the natural tolerance
of the process to the specification. Although there are
many ratios, two, Cp and Cpk are the most widely used and
will be discussed here.
Cp compares the process with the specifications without
regard to centering. It is only concerned with the
variability of the process. It is an indicator of how the
process could do if it was centered perfectly.
Cpk compares the process with both the tolerance and the
centering of the specifications. It is a better indicator
than Cp as to how the process is actually doing than Cp.
The Cp and Cpk of a perfectly centered process would be
exactly the same.
92
The formula for Cp is:
pCtolerance width
6s=
The formula for Cpk is:
⎥⎦
⎤⎢⎣
⎡ −−=
3sXUSL,
3sLSLXminCpk
where s is the estimated standard deviation of the
process.
The data for dot gain from an in-control process were as
follows:
20, 19, 20, 23, 17, 20, 19, 21, 19, 18, 18, 21, 20, 19,
22, 20, 21, 20, 19, 18, 17, 21.
The average and standard deviation of the data are:
X-bar = 19.64
σ = 1.53
The tolerance limits are 18 and 22.
Using the two equations above:
09.153.1*62515
Cp =−
=
[ ] 01.117.1,01.1min53.1*3
64.1925,53.1*3
1564.19minCpk ==⎥⎦⎤
⎢⎣⎡ −−
=
Potential Applications
93
There are many potential applications for Process
Capability ratios. One example would be tracking Cpk over
time to show process improvement. Another potential use
could be tracking supplier quality.
Certain capability ratios are sometimes demanded by
customers. Being able to show a capable process may
increase business opportunities.
Measurement Science
Measurement science is concerned with the correct way of
measuring to ensure accurate and repeatable measurements.
This includes calibration of measuring equipment and
systems, and minimizing variation of those systems.
The principal tool in measurement science is the Gage
Repeatability and Reproducibility (GR&R) study. This study
compares the differences inspectors have in using gages so
the measurement system can be modified to minimize the
variation.
Definitions
Accuracy. Accuracy is the difference between average
readings and the true value.
94
Precision. Precision is often called repeatability. It
refers to the measurement system always producing the same
result.
Reproducibility. Reproducibility refers to different
people obtaining the same results.
Gage Repeatability and Reproducibility
A Gage Repeatability and Reproducibility (GR&R) study is
a method of ensuring that values will be the same over
repeated measurements and with different inspectors. Sample
forms are included in the appendix for use in GR&R studies.
The steps for using the forms are as follows:
1. Choose three inspectors to perform the measurements.
Enter their names as inspectors A, B and C.
2. Each inspector should randomly measure the same ten
parts. This should be repeated again so each inspector has
randomly measured the ten parts twice. It is important the
inspectors cannot differentiate between the parts so a true
measurement is taken. Also, the inspectors should not be
allowed to communicate during the study.
3. Record the difference between the measurements in
the column entitled "RANGE." Then sum and average the
ranges and measurements for each inspector. The X-bar will
be the sum divided by twenty.
95
4. The overall average range can be calculated, along
with the range of the averages. UCLR can be calculated as
3.268 times the average range.
5. The average range and range of the averages can be
carried over to page two. The formulae can be followed to
find the standard variation for repeatability. If there are
three inspectors and ten samples d2 will be 1.13.
6. The variation for reproducibility can be calculated.
With three inspectors the d2 will be 1.91.
7. The combined variation can be calculated as shown.
8. The percent of tolerance used by the measurement
variation can be calculated. If this percentage used is
more than ten percent the measurement system should be
evaluated.
Repeatability refers to equipment variation. If the
repeatability variation is larger than the reproducibility,
the gage should be checked. A faulty gage, or one that is
hard to use and read should be fixed or replaced.
Reproducibility refers to the inspector variation.
Excess reproducibility variation shows the inspectors have
different way of using the gage or reading the gage.
Methods should be established and put in place so each
inspector does it the same way.
After any changes are made to the measurement system
another GR&R study should be done.
96
A GR&R study was to be conducted on quality inspectors
measuring midtone dot gain. Three operators and 10 samples
were gathered to conduct the study. The data and
calculations are on the following pages.
The percent of the tolerance that is consumed by the
GR&R variation is almost 80%. This is far too much. The
calculations show that both repeatability and
reproducibility variation are too big, with reproducibility
the largest. The best course of action would be to acquire
a densitometer with a greater sensitivity, and train the
inspectors on it.
97
GA
GE:
Densitom
eterD
ATE:Fall 1993
SPEC
S:48 - 52
BY:C
PP
PAR
T:C
HAR
ACTER
ISTIC:
Midtone D
ot Gain
SAM
PLE #1ST TR
IAL2N
D TR
IALR
AN
GE
1ST TRIA
L2N
D TR
IALR
ANG
E1S
T TRIAL
2ND
TRIA
LR
ANG
E1
5050
050
491
4949
02
4847
148
491
4748
13
4949
048
491
4849
14
5050
050
491
4951
25
5150
149
512
4950
16
4747
048
480
4847
17
4949
049
481
4949
08
5050
049
501
4849
19
5151
051
510
5050
010
5050
049
501
4950
1Totals:
Sum
:988
0.2985
0.9978
0.849.4
49.2548.9
A-BillB
-TedC
-Jean
GAG
E REPE
ATABILITY AN
D R
EPR
OD
UC
IBILITY
INSP
ECTO
R
633.0
38.
09.
02.
03
=+
+=
++
=R
RR
CB
AR
7.0
=−
=X
XR
MIN
MAX
X
XA
XB
XC
↑R
B↑
RA
↑R
C
98
GAGE REPEATABILITY AND REPRODUCIBILITY
REPEATABILITY (EQUIPMENT VARIATION):
56.013.1633.0
2
===dR
Rσ
d2 factors for repeatability
k 2 3 4 5 6 7 8 9 10 11
d2 1.3 1.2 1.2 1.2 1.2 1.17 1.17 1.16 1.16 1.13
REPRODUCIBILITY (APPRAISER VARIATION):
37.091.17.0
2
===dRX
RXσ
d2 factors for reproducibility
n 2 3
d2 1.4 1.9
REPEATABILITY & REPRODUCIBILITY (COMBINED):
( ) ( ) ( ) ( ) 67.037.056.0 2222
&=+=+= σσσ R XRRR
PERCENT TOLERANCE CONSUMED BY REPEATABILITY AND
REPRODUCIBILITY: ( )( )[ ]
( )( )( )[ ]
( ) %86%1004852
67.015.5%10015.5
... & =−
== XXTOLERANCE
CTP RRσ
99
Conclusion
Process capability can be an important tool for tracking
process improvement over time. Cp and Cpk can show the
response of a process to changes.
Measurement capability is important for the application
of SPC and DOE. Improper measurements invalidate and
studies or experiments undertaken. Gage Repeatability and
Reproducibility studies can minimize measurement variation
and make SPC and DOE more effective.
100
CHAPTER 5: CONCLUSION
The importance of using statistics in industry for the
improvement of quality and productivity is unquestioned.
The printing industry has recently begun to buy into the
quality revolution, but in many cases have not been properly
trained in the use of statistics.
The biggest detraction from the use of statistics is
their ease of misuse. Slight misuse of statistics can cause
improper conclusions to be formulated and wrong decisions to
be made, and can dramatically increase costs to a company.
If this paper has served as an introduction to
statistics for the printer, and stimulated curiosity in
learning more, it has served its purpose. Statistics can be
a boon to any printer willing to learn and apply them
properly, and it is the hope of the author that this will
happen.
101
APPENDIX 1 - TABLES FOR CONTROL LIMITS
Three Sigma X-bar and R Charts
n A2 D3 D4
2 1.880 0 3.267
3 1.023 0 2.574
4 .729 0 2.282
5 .577 0 2.115
6 .483 0 2.004
7 .419 .076 1.924
8 .373 .136 1.864
9 .337 .184 1.816
10 .308 .223 1.777
X-bar Chart R Chart
X=Centerline Centerline = R
UCL = +X RA2 UCL = 4D R
LCL = −X RA2 LCL = 3D R
Three Sigma Individuals Charts
Centerline = X
UCL = +X R2 88.
LCL = −X R2 88.
102
Three Sigma Attribute Charts
p- np-
Centerline = p Centerline = n p
UCL = + −p p p n3 1 / UCL = + −n p n p p3 1
LCL = − −p p p n3 1 / LCL = − −n p n p p3 1
u- c-
Centerline = u Centerline = c
UCL = +u u n3 / UCL = +c c3
LCL = −u u n3 / LCL = +c c3
103
APPENDIX 2 - TAGUCHI EXPERIMENT TABLES
L4Factor Assignment
Main Factors
12
3Interactions
2x31x3
1x21
11
12
12
23
21
24
22
1
L8Factor Assignment
Main Factors
12
34
56
72x3
1x31x2
1x51x4
1x71x6
Interactions4x5
4x64x7
2x62x7
2x42x5
6x75x7
5x63x7
3x63x5
3x41
11
11
11
12
11
12
22
23
12
21
12
24
12
22
21
15
21
21
21
26
21
22
12
17
22
11
22
18
22
12
11
2
L9 - Three LevelFactor Assignm
entM
ain Factors1
23
4Interactions
1x21x2
11
11
12
12
22
31
33
34
21
23
52
23
16
23
12
73
13
28
32
13
93
32
1
104
L12 - No Interactions
Factor Assignm
entM
ain Factors1
23
45
67
89
1011
11
11
11
11
11
11
21
11
11
22
22
22
31
12
22
11
12
22
41
21
22
12
21
12
51
22
12
21
21
21
61
22
21
22
12
11
72
12
21
12
21
21
82
12
12
22
11
12
92
11
22
21
22
11
102
22
11
11
22
12
112
21
21
21
11
22
122
21
12
12
12
21
L16Factor Assignm
entM
ain Factors1
23
45
67
89
1011
1213
1415
2x31x3
1x21x5
1x41x7
1x61x9
1x81x11
1x101x13
1x121x15
1x144x5
4x64x7
2x62x7
2x42x5
2x102x11
2x82x9
2x142x15
2x122x13
6x75x7
5x63x7
3x63x5
3x43x11
3x103x9
3x83x15
3x143x13
3x12Interactions
8x98x10
8x118x12
8x138x14
8x154x12
4x134x14
4x154x8
4x94x10
4x1110x11
9x119x10
9x139x12
9x159x14
5x135x12
5x155x14
5x95x8
5x115x10
12x1312x14
12x1510x14
10x1510x12
10x136x14
6x156x12
6x136x10
6x116x8
6x914x15
13x1513x14
11x1511x14
11x1311x12
7x157x14
7x137x12
7x117x10
7x97x8
11
11
11
11
11
11
11
11
21
11
11
11
22
22
22
22
31
11
22
22
11
11
22
22
41
11
22
22
22
22
11
11
51
22
11
22
11
22
11
22
61
22
11
22
22
11
22
11
71
22
22
11
11
22
22
11
81
22
22
11
22
11
11
22
92
12
12
12
12
12
12
12
102
12
12
12
21
21
21
21
112
12
21
21
12
12
21
21
122
12
21
21
21
21
12
12
132
21
12
21
12
21
12
21
142
21
12
21
21
12
21
12
152
21
21
12
12
21
21
12
162
21
21
12
21
12
12
21
105
L27 - Three LevelFactor Assignm
entM
ain Factors1
23
45
67
89
1011
1213
2x31x3
1x21x2
1x61x5
1x51x9
1x81x8
1x121x11
1x112x4
1x41x4
1x31x7
1x71x6
1x101x9
1x91x13
1x131x12
3x43x4
2x42x3
2x82x9
2x102x5
2x62x7
2x52x6
2x75x6
5x85x9
5x102x11
2x122x13
2x112x12
2x132x8
2x92x10
5x75x11
5x135x12
3x93x10
3x83x7
3x53x6
3x63x7
3x5Interactions
6x76x9
6x106x8
3x133x11
3x123x12
3x133x11
3x103x8
3x98x9
6x126x11
6x134x10
4x84x9
4x64x7
4x54x7
4x54x6
8x107x10
7x87x9
4x124x13
4x114x13
4x114x12
4x94x10
4x89x10
7x137x12
7x116x7
5x75x6
5x115x13
5x125x8
5x105x9
11x128x11
8x128x13
8x118x13
8x126x13
6x126x11
6x106x9
6x811x13
9x129x13
9x119x13
9x129x11
7x127x11
7x137x9
7x87x10
12x1310x13
10x1310x12
10x1210x11
10x139x10
8x108x9
12x1311x13
11x121
11
11
11
11
11
11
12
11
11
22
22
22
22
23
11
11
33
33
33
33
34
12
22
11
12
22
33
35
12
22
22
23
33
11
16
12
22
33
31
11
22
27
13
33
11
13
33
22
28
13
33
22
21
11
33
39
13
33
33
32
22
11
110
21
23
12
31
23
12
311
21
23
23
12
31
23
112
21
23
31
23
12
31
213
22
31
12
32
31
31
214
22
31
23
13
12
12
315
22
31
31
21
23
23
116
23
12
12
33
12
23
117
23
12
23
11
23
31
218
23
12
31
22
31
12
319
31
32
13
21
32
13
220
31
32
21
32
13
21
321
31
32
32
13
21
32
122
32
13
13
22
13
32
123
32
13
21
33
21
13
224
32
13
32
11
32
21
325
33
21
13
23
21
21
326
11
11
11
11
11
11
127
33
21
32
12
13
13
2
106
APPENDIX 3 - GR&R FORMS
GA
GE:
DATE:
SPECS:
BY:
PAR
T:C
HAR
ACTER
ISTIC:
SAMPLE #
1ST TRIAL
2ND
TRIAL
RAN
GE
1ST TRIAL
2ND
TRIAL
RAN
GE
1ST TRIAL
2ND
TRIAL
RAN
GE
12345678910Totals:
Sum:
AB
C
GAG
E REPEATABILITY AN
D R
EPR
OD
UC
IBILITY
INSPEC
TOR
=+
+=
++
=3
3R
RR
CB
AR
=−
=X
XR
MIN
MAX
X
XA
XB
XC
↑R
B↑
RA
↑R
C
107
GAGE REPEATABILITY AND REPRODUCIBILITY
REPEATABILITY (EQUIPMENT VARIATION):
===dR
R2
σ
d2 factors for repeatability
k 2 3 4 5 6 7 8 9 10 11
d2 1.3 1.2 1.2 1.2 1.2 1.17 1.17 1.16 1.16 1.13
REPRODUCIBILITY (APPRAISER VARIATION):
===dRX
RX2
σ
d2 factors for reproducibility
n 2 3
d2 1.4 1.9
REPEATABILITY & REPRODUCIBILITY (COMBINED):
( ) ( ) ( ) ( ) =+=+=2222
& σσσ R XRRR
PERCENT TOLERANCE CONSUMED BY REPEATABILITY AND
REPRODUCIBILITY: ( )( )[ ]
( )( )( )[ ]
( ) === %10015.5%10015.5
... & XXTOLERANCE
CTP RRσ
108
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Finn, L., Kramer, T. & Reynard, S. (1987). Design of
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Walpole, R.E. & Myers, R.H. (1989) Probability and
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