1 Statistical Quality Control, 7th Edition by Douglas C. Montgomery

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
1
Copyright (c) 2013 John Wiley & Sons, Inc.
Basic SPC Tools
Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 2

5.2 Chance and Assignable Causes of Variation
• A process is operating with only chance causes of variation present is said to
be in statistical control.
• A process that is operating in the presence of assignable causes is said to be
out of control.

5.3 Statistical Basis of the Control Chart
• A control chart contains

– A center line
– An upper control limit
– A lower control limit
• A point that plots within the
control limits indicates the
process is in control
– No action is necessary
• A point that plots outside the
control limits is evidence that the
process is out of control
– Investigation and corrective
action are required to find and
eliminate assignable cause(s)
• There is a close connection
between control charts and
hypothesis testing

Photolithography Example
• Important quality characteristic in hard bake is resist flow width

• Process is monitored by average flow width
– Sample of 5 wafers
– Process mean is 1.5 microns
– Process standard deviation is 0.15 microns
• Note that all plotted points fall inside the control limits
– Process is considered to be in statistical control

Determination of the Control Limits

Shewhart Control Chart Model

How the Shewhart Control Chart Works

Out-Of-Control-Action Plans

More Basic Principles
• Control charts may be used to estimate process
parameters, which are used to determine capability
• Two general types of control charts
– Variables (Chapter 6)
• Continuous scale of measurement
• Quality characteristic described by central tendency and a measure
of variability
– Attributes (Chapter 7)
• Conforming/nonconforming
• Counts
• Control chart design encompasses selection of
sample size, control limits, and sampling frequency

Types of Process Variability
• Stationary and uncorrelated − data vary around a fixed mean in a stable
or predictable manner
• Stationary and autocorrelated − successive observations are dependent
with tendency to move in long runs on either side of mean
• Nonstationary − process drifts without any sense of a stable or fixed mean

Reasons for Popularity
of Control Charts
1. Control charts are a proven technique for improving
productivity.
2. Control charts are effective in defect prevention.
3. Control charts prevent unnecessary process
adjustment.
4. Control charts provide diagnostic information.
5. Control charts provide information about process
capability.

• 3-Sigma Control Limits
– Probability of type I error is 0.0027
• Probability Limits
– Type I error probability is chosen directly
– For example, 0.001 gives 3.09-sigma control limits
• Warning Limits
– Typically selected as 2-sigma limits

5.3.3 Sample Size and Sampling Frequency

LSL USL
Chapter 5 Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 20
Consider the hard-bake process discussed earlier, and suppose we are sampling
every hour. Equation 5.3 indicates that we will have a false alarm about every
370 hours on the average. Now consider how the control chart performs in
detecting shifts in the mean.
Suppose we are using a sample size of n = 5 and that when the process goes out
of control the mean shifts to 1.65 microns. From the operating characteristic
curve in Figure 5.9 we find that if the process mean is 1.65 microns, the
probability of falling between the control limits is approximately 0.75. Therefore, p
in equation 5.2 is 0.25, and the out-of-control ARL (called ARL1) is
ARL1=1/p=1/0.25=4
That is, the control chart will require 1.54 samples to detect the process shift, on
the average, and since the time interval between samples is h = 1 hour, the
average time required to detect this shift is
ATS = ARL1h=4 hours

Suppose that this is unacceptable, because production of wafers with mean flow
width of 1.65 microns results in excessive scrap costs and can result in further
upstream manufacturing problems.
How can we reduce the time needed to detect the out-of-control condition?
One method is to sample more frequently. For example, if we sample every half
hour, then the average time to signal for this scheme is
ATS = ARL1 h = 4(1/2) = 2
only 2 hours will elapse (on the average) between the shift and its detection. The
second possibility is to increase the sample size. For example, if we use n = 10,
then Figure 5.9 shows that the probability of falling between the control limits
when the process mean is 1.65 microns is approximately 0.49, so that p = 0.51,
and from equation 5.2 the out-of-control ARL or ARL1 is and, if we sample every
hour, the average time to signal is
ARL1=1/p=1/0.51=1.96, ATS = ARL1 h = 1.96*1=1.96 hours
Thus, the larger sample size would allow the shift to be detected more quickly
than with the smaller one.

5.3.4 Rational Subgroups
• The rational subgroup concept means that subgroups or samples
should be selected so that if assignable causes are present, chance for
differences between subgroups will be maximized, while chance for
difference due to assignable causes within a subgroup will be
minimized.
• Two general approaches for constructing rational subgroups:

1. Sample consists of units produced at the same time − consecutive units
– Primary purpose is to detect process shifts
2. Sample consists of units that are representative of all units produced
since last sample − random sample of all process output over
sampling interval
– Often used to make decisions about acceptance of product
– Effective at detecting shifts to out-of-control state and back into in-control
state between samples
– Care must be taken because we can often make any process appear to be
in statistical control just by stretching out the interval between
observations in the sample.

Rational Subgroups

5.3.5 Patterns on Control Charts
• Pattern is very nonrandom in appearance

• 19 of 25 points plot below the center line, while only 6 plot
above
• Following 4th point, 5 points in a row increase in
magnitude, a run up
• There is also an unusually long run down beginning with
18th point
The Cyclic Pattern

5.3.6 Discussion of the Sensitizing Rules

See Champ and Woodall (1987)

4.3.7 Phase I and Phase II of Control Chart Application
• Phase I is a retrospective analysis of process

data to construct trial control limits
– Charts are effective at detecting large, sustained
shifts in process parameters, outliers, measurement
errors, data entry errors, etc.
– Facilitates identification and removal of assignable
causes
• In phase II, the control chart is used to monitor
the process
– Process is assumed to be reasonably stable
– Emphasis is on process monitoring, not on bringing
an unruly process into control

5.4 THE REST OF THE “MAGNIFICENT SEVEN”
1. Histogram or stem-and-leaf plot

2. Check sheet
3. Pareto chart
4. Cause-and-effect diagram
5. Defect concentration diagram
6. Scatter diagram
7. Control chart

Check Sheet

Pareto Chart

Cause-and-Effect Diagram

Defect Concentration Diagram

Scatter Diagram

5.5 Implementing SPC in a Quality Improvement Program

5.6 An Application of SPC
• Improving quality in a copper plating operation at a
printed circuit board fabrication plant
• The DMAIC process was used
• During the define step, the team decided to focus on
reducing flow time through the process
• During the measures step, controller downtown was
recognized as a major factor in excessive flow time

5.7 Applications of SPC and Quality Improvement
Tools in Transactional and Service Businesses
• Nonmanufacturing applications often do not differ
substantially from industrial applications, but
sometimes require ingenuity
1. Most nonmanufacturing operations do not have a natural
measurement system
2. The observability of the process may be fairly low
3. People are usually involved in transactional and services
processes, and variability between people may be an
important part of the problem
• Flow charts, operation process charts and value
stream mapping are particularly useful in developing
process definition and process understanding. This is
sometimes called process mapping.
– Used to identify value-added versus nonvalue-added
activity

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Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery.

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 2

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 3

• A control chart contains

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 4

• Important quality characteristic in hard bake is resist flow width

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Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 7

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 8

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Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 11

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Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 15

ATS = ARL1h=4 hours

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 22

ATS = ARL1 h = 4(1/2) = 2

ARL1=1/p=1/0.51=1.96, ATS = ARL1 h = 1.96*1=1.96 hours

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 24

• Two general approaches for constructing rational subgroups:

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 26

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 27

• Pattern is very nonrandom in appearance

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 29

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 31

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 32

• Phase I is a retrospective analysis of process

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 34

1. Histogram or stem-and-leaf plot

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 35

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 36

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 37

Chapter 5 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 39

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