Chapter 4
Statistical Process Control
Operations Management - 5th Edition
Roberta Russell & Bernard W. Taylor, III
Copyright 2006 John Wiley & Sons, Inc.
Beni Asllani
University of Tennessee at Chattanooga
Lecture Outline







Basics of Statistical Process Control
Control Charts
Control Charts for Attributes
Control Charts for Variables
Control Chart Patterns
SPC with Excel
Process Capability
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4-2
Basics of Statistical
Process Control
 Statistical Process Control
(SPC)

monitoring production process
to detect and prevent poor
quality
UCL
 Sample

subset of items produced to
use for inspection
LCL
 Control Charts

process is within statistical
control limits
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4-3
Variability
 Random



common causes
inherent in a process
can be eliminated
only through
improvements in the
system
Copyright 2006 John Wiley & Sons, Inc.
 Non-Random



special causes
due to identifiable
factors
can be modified
through operator or
management action
4-4
SPC in TQM
 SPC


tool for identifying problems and
make improvements
contributes to the TQM goal of
continuous improvements
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4-5
Quality Measures
 Attribute


a product characteristic that can be
evaluated with a discrete response
good – bad; yes - no
 Variable


a product characteristic that is continuous
and can be measured
weight - length
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4-6
Applying SPC to
Service
 Nature of defect is different in services
 Service defect is a failure to meet
customer requirements
 Monitor times, customer satisfaction
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4-7
Applying SPC to
Service (cont.)
 Hospitals

timeliness and quickness of care, staff responses to requests,
accuracy of lab tests, cleanliness, courtesy, accuracy of
paperwork, speed of admittance and checkouts
 Grocery Stores

waiting time to check out, frequency of out-of-stock items,
quality of food items, cleanliness, customer complaints,
checkout register errors
 Airlines

flight delays, lost luggage and luggage handling, waiting time
at ticket counters and check-in, agent and flight attendant
courtesy, accurate flight information, passenger cabin
cleanliness and maintenance
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4-8
Applying SPC to
Service (cont.)
 Fast-Food Restaurants

waiting time for service, customer complaints,
cleanliness, food quality, order accuracy, employee
courtesy
 Catalogue-Order Companies

order accuracy, operator knowledge and courtesy,
packaging, delivery time, phone order waiting time
 Insurance Companies

billing accuracy, timeliness of claims processing,
agent availability and response time
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4-9
Where to Use Control Charts
 Process has a tendency to go out of control
 Process is particularly harmful and costly if it
goes out of control
 Examples




at the beginning of a process because it is a waste of
time and money to begin production process with bad
supplies
before a costly or irreversible point, after which
product is difficult to rework or correct
before and after assembly or painting operations that
might cover defects
before the outgoing final product or service is
delivered
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Control Charts
 A graph that establishes
control limits of a
process
 Control limits

 Types of charts

Attributes
p-chart
 c-chart

upper and lower bands of
a control chart

Variables
range (R-chart)
 mean (x bar – chart)

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Process Control
Chart
Out of control
Upper
control
limit
Process
average
Lower
control
limit
1
2
3
4
5
6
7
8
9
10
Sample number
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4-12
Normal Distribution
95%
99.74%
-3
-2
-1
Copyright 2006 John Wiley & Sons, Inc.
=0
1
2
3
4-13
A Process Is in
Control If …
1. … no sample points outside limits
2. … most points near process average
3. … about equal number of points above
and below centerline
4. … points appear randomly distributed
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Control Charts for
Attributes
 p-charts
 uses portion defective in a sample
 c-charts
 uses number of defects in an item
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4-15
p-Chart
UCL = p + zp
LCL = p - zp
z = number of standard deviations from
process average
p = sample proportion defective; an estimate
of process average
p = standard deviation of sample proportion
p =
Copyright 2006 John Wiley & Sons, Inc.
p(1 - p)
n
4-16
p-Chart Example
SAMPLE
NUMBER OF
DEFECTIVES
PROPORTION
DEFECTIVE
6
0
4
:
:
18
200
.06
.00
.04
:
:
.18
1
2
3
:
:
20
20 samples of 100 pairs of jeans
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p-Chart Example (cont.)
total defectives
p = total sample observations = 200 / 20(100) = 0.10
UCL = p + z
p(1 - p)
= 0.10 + 3
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = p - z
p(1 - p)
= 0.10 - 3
n
0.10(1 - 0.10)
100
LCL = 0.010
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0.20
UCL = 0.190
0.18
p-Chart
Example
(cont.)
Proportion defective
0.16
0.14
0.12
0.10
p = 0.10
0.08
0.06
0.04
0.02
LCL = 0.010
2
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4
6
8
10
12 14
Sample number
16
18
20
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c-Chart
UCL = c + zc
LCL = c - zc
c =
c
where
c = number of defects per sample
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c-Chart (cont.)
Number of defects in 15 sample rooms
SAMPLE
NUMBER
OF
DEFECTS
1
2
3
12
8
16
:
:
:
:
15
15
190
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190
c=
= 12.67
15
UCL = c + zc
= 12.67 + 3
= 23.35
12.67
LCL = c + zc
= 12.67 - 3
= 1.99
12.67
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24
UCL = 23.35
c-Chart
(cont.)
Number of defects
21
18
c = 12.67
15
12
9
6
LCL = 1.99
3
2
4
6
8
10
12
14
16
Sample number
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Control Charts for
Variables
 Mean chart ( x -Chart )
 uses average of a sample
 Range chart ( R-Chart )
 uses amount of dispersion in a
sample
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4-23
x-bar Chart
x1 + x2 + ... xk
=
x=
k
=
UCL = x + A2R
=
LCL = x - A2R
where
=
x = average of sample means
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4-24
x-bar Chart Example
OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k
1
2
3
4
5
x
R
1
2
3
4
5
6
7
8
9
10
5.02
5.01
4.99
5.03
4.95
4.97
5.05
5.09
5.14
5.01
5.01
5.03
5.00
4.91
4.92
5.06
5.01
5.10
5.10
4.98
4.94
5.07
4.93
5.01
5.03
5.06
5.10
5.00
4.99
5.08
4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99
4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
50.09
1.15
Example 15.4
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4-25
x- bar Chart
Example (cont.)
50.09
= x
x=
=
= 5.01 cm
k
10
UCL = x= + A2R = 5.01 + (0.58)(0.115) = 5.08
LCL = x= - A2R = 5.01 - (0.58)(0.115) = 4.94
Retrieve Factor Value A2
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4-26
5.10 –
5.08 –
UCL = 5.08
5.06 –
Mean
5.04 –
x- bar
Chart
Example
(cont.)
5.02 –
x= = 5.01
5.00 –
4.98 –
4.96 –
LCL = 4.94
4.94 –
4.92 –
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Sample number
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R- Chart
UCL = D4R
LCL = D3R
R
R= k
where
R = range of each sample
k = number of samples
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R-Chart Example
OBSERVATIONS (SLIP-RING DIAMETER, CM)
SAMPLE k
1
2
3
4
5
x
R
1
2
3
4
5
6
7
8
9
10
5.02
5.01
4.99
5.03
4.95
4.97
5.05
5.09
5.14
5.01
5.01
5.03
5.00
4.91
4.92
5.06
5.01
5.10
5.10
4.98
4.94
5.07
4.93
5.01
5.03
5.06
5.10
5.00
4.99
5.08
4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99
4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
50.09
1.15
Example 15.3
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4-29
R-Chart Example (cont.)
R
1.15
R=
=
= 0.115
k
10
UCL = D4R = 2.11(0.115) = 0.243
LCL = D3R = 0(0.115) = 0
Retrieve Factor Values D3 and D4
Example 15.3
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R-Chart Example (cont.)
0.28 –
0.24 –
Range
0.20 –
0.16 –
UCL = 0.243
R = 0.115
0.12 –
0.08 –
0.04 –
0–
LCL = 0
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Using x- bar and R-Charts
Together
 Process average and process variability must be
in control.
 It is possible for samples to have very narrow
ranges, but their averages is beyond control
limits.
 It is possible for sample averages to be in
control, but ranges might be very large.
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4-32
Control Chart Patterns
UCL
UCL
LCL
Sample observations
consistently below the
center line
LCL
Sample observations
consistently above the
center line
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4-33
Control Chart Patterns (cont.)
UCL
UCL
LCL
Sample observations
consistently increasing
LCL
Sample observations
consistently decreasing
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4-34
Zones for Pattern Tests
=
3 sigma = x + A2R
UCL
Zone A
= 2
2 sigma = x + 3 (A2R)
Zone B
= 1
1 sigma = x + 3 (A2R)
Zone C
=
x
Process
average
Zone C
=
1 sigma = x - 1 (A2R)
3
Zone B
=
2 sigma = x - 2 (A2R)
3
Zone A
=
3 sigma = x - A2R
LCL
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Sample number
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Control Chart Patterns





8 consecutive points on one side of the center line
8 consecutive points up or down across zones
14 points alternating up or down
2 out of 3 consecutive points in zone A but still
inside the control limits
4 out of 5 consecutive points in zone A or B
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4-36
Performing a Pattern Test
SAMPLE
1
2
3
4
5
6
7
8
9
10
x
ABOVE/BELOW
UP/DOWN
ZONE
4.98
5.00
4.95
4.96
4.99
5.01
5.02
5.05
5.08
5.03
B
B
B
B
B
—
A
A
A
A
—
U
D
D
U
U
U
U
U
D
B
C
A
A
C
C
C
B
A
B
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4-37
Sample Size
 Attribute charts require larger sample sizes
 50 to 100 parts in a sample
 Variable charts require smaller samples
 2 to 10 parts in a sample
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4-38
SPC with Excel
UCL=0.19
LCL=0.01
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4-39
SPC with Excel:
Formulas
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4-40
Process Capability
 Tolerances

design specifications reflecting product
requirements
 Process capability

range of natural variability in a process what
we measure with control charts
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4-41
Process Capability
Design
Specifications
(a) Natural variation
exceeds design
specifications; process
is not capable of
meeting specifications
all the time.
Process
Design
Specifications
(b) Design specifications
and natural variation the
same; process is capable
of meeting specifications
most of the time.
Process
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4-42
Process Capability (cont.)
Design
Specifications
(c) Design specifications
greater than natural
variation; process is
capable of always
conforming to
specifications.
Process
Design
Specifications
(d) Specifications greater
than natural variation,
but process off center;
capable but some output
will not meet upper
specification.
Process
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4-43
Process Capability Measures
Process Capability Ratio
Cp =
tolerance range
process range
upper specification limit lower specification limit
=
6
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4-44
Computing Cp
Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
upper specification limit lower specification limit
Cp =
6
9.5 - 8.5
=
= 1.39
6(0.12)
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4-45
Process Capability Measures
Process Capability Index
Cpk = minimum
=
x - lower specification limit
,
3
=
upper specification limit - x
3
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4-46
Computing Cpk
Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
Cpk = minimum
= minimum
=
x - lower specification limit
,
3
=
upper specification limit - x
3
8.80 - 8.50 9.50 - 8.80
,
= 0.83
3(0.12)
3(0.12)
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4-47
Appendix:
Determining Control Limits for x-bar and R-Charts
SAMPLE SIZE
n
FACTOR FOR x-CHART
A2
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.44
0.11
0.99
0.77
0.55
0.44
0.22
0.11
0.00
0.99
0.99
0.88
Fact
ors
Copyright 2006 John Wiley & Sons, Inc.
Return
FACTORS FOR R-CHART
D3
D4
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
0.36
0.38
0.39
0.40
0.41
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
1.64
1.62
1.61
1.61
1.59
4-48
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