statistical process control 1 chapter 3. lecture outline basics of statistical process control...
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Copyright 2011 John Wiley & Sons, Inc.
Lecture Outline
• Basics of Statistical Process Control• Control Charts• Control Charts for Attributes• Control Charts for Variables• Control Chart Patterns• SPC with Excel and OM Tools• Process Capability
3-2
Copyright 2011 John Wiley & Sons, Inc.
Statistical Process Control (SPC)
• Statistical Process Control• monitoring production process
to detect and prevent poor quality
• Sample• subset of items produced to
use for inspection
• Control Charts• process is within statistical
control limits
3-3
UCL
LCL
Copyright 2011 John Wiley & Sons, Inc.
Process Variability
• Random• inherent in a process• depends on equipment
and machinery, engineering, operator, and system of measurement
• natural occurrences
• Non-Random• special causes• identifiable and
correctable• include equipment out of
adjustment, defective materials, changes in parts or materials, broken machinery or equipment, operator fatigue or poor work methods, or errors due to lack of training
3-4
Copyright 2011 John Wiley & Sons, Inc.
SPC in Quality Management
• SPC uses• Is the process in control?• Identify problems in order to make
improvements• Contribute to the TQM goal of continuous
improvement
3-5
Copyright 2011 John Wiley & Sons, Inc.
Quality Measures:Attributes and Variables
• Attribute• A characteristic which is evaluated with a
discrete response• good/bad; yes/no; correct/incorrect
• Variable measure• A characteristic that is continuous and can be
measured• Weight, length, voltage, volume
3-6
Copyright 2011 John Wiley & Sons, Inc.
SPC Applied to Services
• Nature of defects is different in services• Service defect is a failure to meet customer
requirements• Monitor time and customer satisfaction
3-7
Copyright 2011 John Wiley & Sons, Inc.
SPC Applied to Services
• Hospitals• timeliness & quickness of care, staff responses to requests,
accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance & 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 & luggage handling, waiting time at
ticket counters & check-in, agent & flight attendant courtesy, accurate flight information, cabin cleanliness & maintenance
3-8
Copyright 2011 John Wiley & Sons, Inc.
SPC Applied to Services
• Fast-food restaurants• waiting time for service, customer complaints, cleanliness, food
quality, order accuracy, employee courtesy
• Catalogue-order companies• order accuracy, operator knowledge & courtesy, packaging,
delivery time, phone order waiting time
• Insurance companies• billing accuracy, timeliness of claims processing, agent
availability & response time
3-9
Copyright 2011 John Wiley & Sons, Inc.
Where to Use Control Charts
• Process • Has a tendency to go out of control
• Is particularly harmful and costly if it goes out of control
• Examples• At beginning of process because of waste 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
3-10
Copyright 2011 John Wiley & Sons, Inc.
Control Charts
• A graph that monitors process quality• Control limits
• upper and lower bands of a control chart
• Attributes chart• p-chart• c-chart
• Variables chart• mean (x bar – chart)• range (R-chart)
3-11
Copyright 2011 John Wiley & Sons, Inc.
Process Control Chart
3-12
1 2 3 4 5 6 7 8 9 10Sample number
Uppercontrol
limit
Processaverage
Lowercontrol
limit
Out of control
Copyright 2011 John Wiley & Sons, Inc.
Normal Distribution
• Probabilities for Z= 2.00 and Z = 3.00
3-13
=0 1 2 3-1-2-3
95%
99.74%
Copyright 2011 John Wiley & Sons, Inc.
A Process Is in Control If …
3-14
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
Copyright 2011 John Wiley & Sons, Inc.
Control Charts for Attributes
• p-chart• uses portion defective in a sample
• c-chart• uses number of defects (non-conformities) in a
sample
3-15
Copyright 2011 John Wiley & Sons, Inc.
p-Chart
3-16
UCL = p + zp
LCL = p - zp
z = number of standard deviations from process average
p = sample proportion defective; estimates process mean
p = standard deviation of sample proportionp =
p(1 - p)
n
Copyright 2011 John Wiley & Sons, Inc.
Construction of p-Chart
3-17
20 samples of 100 pairs of jeans
NUMBER OF PROPORTIONSAMPLE # DEFECTIVES DEFECTIVE
1 6 .06
2 0 .00
3 4 .04
: : :
: : :
20 18 .18
200
Copyright 2011 John Wiley & Sons, Inc.
Construction of p-Chart
3-18
UCL = p + z = 0.10 + 3p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10total defectives
total sample observationsp =
Copyright 2011 John Wiley & Sons, Inc.
Construction of p-Chart
3-19
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Pro
por
tion
defe
ctiv
e
Sample number2 4 6 8 10 12 14 16 18 20
UCL = 0.190
LCL = 0.010
p = 0.10
Copyright 2011 John Wiley & Sons, Inc.
p-Chart in Excel
3-20
Click on “Insert” then “Charts”
to construct control chart
I4 + 3*SQRT(I4*(1-I4)/100)
I4 - 3*SQRT(I4*(1-I4)/100)
Column values copiedfrom I5 and I6
Copyright 2011 John Wiley & Sons, Inc.
c-Chart
3-21
UCL = c + zc
LCL = c - zc
where
c = number of defects per sample
c = c
Copyright 2011 John Wiley & Sons, Inc.
c-Chart
3-22
Number of defects in 15 sample rooms
1 122 83 16
: :: :15 15 190
SAMPLE
c = = 12.6719015
UCL = c + zc
= 12.67 + 3 12.67= 23.35
LCL = c - zc
= 12.67 - 3 12.67= 1.99
NUMBER OF
DEFECTS
Copyright 2011 John Wiley & Sons, Inc.
c-Chart
3-23
3
6
9
12
15
18
21
24
Num
ber
of
defe
cts
Sample number
2 4 6 8 10 12 14 16
UCL = 23.35
LCL = 1.99
c = 12.67
Copyright 2011 John Wiley & Sons, Inc.
Control Charts for Variables
3-24
Range chart ( R-Chart ) Plot sample range (variability)
Mean chart ( x -Chart ) Plot sample averages
3-25
x-bar Chart: Known
UCL = x + z x LCL = x - z x
-
-
==
Where
s = process standard deviations
x = standard deviation of sample means =/k = number of samples (subgroups)n = sample size (number of observations)
x1 + x2 + ... + xk
kX = =
- - -
n
Copyright 2011 John Wiley & Sons, Inc.
Copyright 2011 John Wiley & Sons, Inc.
Observations(Slip-Ring Diameter, cm) n
Sample k 1 2 3 4 5 -
x-bar Chart Example: Known
3-26
x
We know σ = .08
x-bar Chart Example: Known
Copyright 2011 John Wiley & Sons, Inc. 3-27
= 5.01 - 3(.08 / )10
= 4.93
_____10
50.09 = 5.01X =
=
LCL = x - z x = -UCL = x + z x
= -= 5.01 + 3(.08 / )10
= 5.09
Copyright 2011 John Wiley & Sons, Inc.
x-bar Chart Example: Unknown
3-28
_UCL = x + A2R LCL = x - A2R
= =_
where
x = average of the sample means
R = average range value
=_
Copyright 2011 John Wiley & Sons, Inc. 3-29
Control Chart Factors
n A2 D3 D42 1.880 0.000 3.2673 1.023 0.000 2.5754 0.729 0.000 2.2825 0.577 0.000 2.1146 0.483 0.000 2.0047 0.419 0.076 1.9248 0.373 0.136 1.8649 0.337 0.184 1.81610 0.308 0.223 1.77711 0.285 0.256 1.74412 0.266 0.283 1.71713 0.249 0.307 1.69314 0.235 0.328 1.67215 0.223 0.347 1.65316 0.212 0.363 1.63717 0.203 0.378 1.62218 0.194 0.391 1.60919 0.187 0.404 1.59620 0.180 0.415 1.58521 0.173 0.425 1.57522 0.167 0.435 1.56523 0.162 0.443 1.55724 0.157 0.452 1.54825 0.153 0.459 1.541
Factors for R-chartSample
Size Factor for X-chart
Copyright 2011 John Wiley & Sons, Inc.
x-bar Chart Example: Unknown
3-30
OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15Totals
Copyright 2011 John Wiley & Sons, Inc.
x-bar Chart Example: Unknown
3-31
∑ Rk
1.1510R = = = 0.115
_ ____ ____
_UCL = x + A2R
=
= 50.0910
_____x = = = 5.01 cmåx
k___
= 5.01 + (0.58)(0.115) = 5.08
_LCL = x - A2R
== 5.01 - (0.58)(0.115) = 4.94
Copyright 2011 John Wiley & Sons, Inc.
x- bar Chart
Example
3-32
UCL = 5.08
LCL = 4.94
Mea
n
Sample number
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01=
Copyright 2011 John Wiley & Sons, Inc.
R- Chart
3-33
UCL = D4R LCL = D3R
R = åRk
WhereR = range of each samplek = number of samples (sub groups)
Copyright 2011 John Wiley & Sons, Inc.
R-Chart Example
3-34
OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15Totals
Copyright 2011 John Wiley & Sons, Inc.
R-Chart Example
3-35
Retrieve chart factors D3 and D4
UCL = D4R = 2.11(0.115) = 0.243
LCL = D3R = 0(0.115) = 0_
_
Copyright 2011 John Wiley & Sons, Inc.
R-Chart Example
3-36
UCL = 0.243
LCL = 0
Ra
nge
Sample number
R = 0.115
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
0.28 –
0.24 –
0.20 –
0.16 –
0.12 –
0.08 –
0.04 –
0 –
Copyright 2011 John Wiley & Sons, Inc.
Using x- bar and R-Charts Together
• Process average and process variability must be in control
• Samples can have very narrow ranges, but sample averages might be beyond control limits
• Or, sample averages may be in control, but ranges might be out of control
• An R-chart might show a distinct downward trend, suggesting some nonrandom cause is reducing variation
3-38
Copyright 2011 John Wiley & Sons, Inc.
Control Chart Patterns
• Run• sequence of sample values that display same
characteristic
• Pattern test• determines if observations within limits of a control
chart display a nonrandom pattern
3-39
Copyright 2011 John Wiley & Sons, Inc.
Control Chart Patterns
• To identify a pattern look for:• 8 consecutive points on one side of the center line• 8 consecutive points up or down• 14 points alternating up or down• 2 out of 3 consecutive points in zone A (on one side of
center line)• 4 out of 5 consecutive points in zone A or B (on one
side of center line)
3-40
Copyright 2011 John Wiley & Sons, Inc.
Control Chart Patterns
3-41
UCL
LCL
Sample observationsconsistently above thecenter line
LCL
UCL
Sample observationsconsistently below thecenter line
Copyright 2011 John Wiley & Sons, Inc.
Control Chart Patterns
3-42
LCL
UCL
Sample observationsconsistently increasing
UCL
LCL
Sample observationsconsistently decreasing
Zones for Pattern Tests
3-43
UCL
LCL
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Process average
3 sigma = x + A2R=
3 sigma = x - A2R=
2 sigma = x + (A2R)= 23
2 sigma = x - (A2R)= 23
1 sigma = x + (A2R)= 13
1 sigma = x - (A2R)= 13
x=
Sample number
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
Copyright 2011 John Wiley & Sons, Inc.
Copyright 2011 John Wiley & Sons, Inc.
Performing a Pattern Test
3-44
1 4.98 B — B
2 5.00 B U C
3 4.95 B D A
4 4.96 B D A
5 4.99 B U C
6 5.01 — U C
7 5.02 A U C
8 5.05 A U B
9 5.08 A U A
10 5.03 A D B
SAMPLE x ABOVE/BELOW UP/DOWN ZONE
Copyright 2011 John Wiley & Sons, Inc.
Sample Size Determination
• 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
3-45
Copyright 2011 John Wiley & Sons, Inc.
Process Capability
• Compare natural variability to design variability • Natural variability
• What we measure with control charts• Process mean = 8.80 oz, Std dev. = 0.12 oz
• Tolerances• Design specifications reflecting product
requirements• Net weight = 9.0 oz 0.5 oz • Tolerances are 0.5 oz
3-46
Copyright 2011 John Wiley & Sons, Inc.
Process Capability
3-47
(b) Design specifications and natural variation the same; process is capable of meeting specifications most of the time.
Design Specifications
Process
(a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time.
Design Specifications
Process
Copyright 2011 John Wiley & Sons, Inc.
Process Capability
3-48
(c) Design specifications greater than natural variation; process is capable of always conforming to specifications.
Design Specifications
Process
(d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification.
Design Specifications
Process
Copyright 2011 John Wiley & Sons, Inc.
Process Capability Ratio
3-49
Cp =tolerance range
process range
upper spec limit - lower spec limit
6=
Copyright 2011 John Wiley & Sons, Inc.
Computing Cp
3-50
Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz
Cp =
= = 1.39
upper specification limit - lower specification limit
6
9.5 - 8.5
6(0.12)
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Process Capability Index
3-51
Cpk = minimum
x - lower specification limit
3
=
upper specification limit - x
3
=
,
Copyright 2011 John Wiley & Sons, Inc.
Computing Cpk
3-52
Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz
Cpk = minimum
= minimum , = 0.83
x - lower specification limit
3
=
upper specification limit - x
3
=
,
8.80 - 8.50
3(0.12)
9.50 - 8.80
3(0.12)
Copyright 2011 John Wiley & Sons, Inc.
Process Capability With Excel
3-53
=(D6-D7)/(6*D8)
See formula bar
Copyright 2011 John Wiley & Sons, Inc. 3-55
Copyright 2011 John Wiley & Sons, Inc.All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein.
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