reid & sanders, operations management © wiley 2002 statistical quality control 6 c h a p t e r
TRANSCRIPT
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Reid & Sanders, Operations Management© Wiley 2002
Statistical Quality Control 6C H A P T E R
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Page 2Reid & Sanders, Operations Management© Wiley 2002
Learning Objectives
• Describe quality control methods• Understand the use of statistical process
control • Describe & apply control charts• Distinguish x-bar, R, p and c-charts• Define process capability • Describe & apply capability indexes• Define six-sigma capability
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Page 3Reid & Sanders, Operations Management© Wiley 2002
Quality Control Methods
• Descriptive statistics:– Used to describe distributions of data
• Statistical process control (SPC):– Used to determine whether a process is
performing as expected
• Acceptance sampling:– Used to accept or reject entire batches by
only inspecting a few items
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Page 4Reid & Sanders, Operations Management© Wiley 2002
Descriptive Statistics
• Mean (x-bar):– The average or central tendency of a data set
• Standard deviation (sigma):– Describes the amount of spread or observed
variation in the data set
• Range:– Another measure of spread – The range measures the difference between the
largest & smallest observed values in the data set
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Page 5Reid & Sanders, Operations Management© Wiley 2002
The Normal Distribution
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Page 6Reid & Sanders, Operations Management© Wiley 2002
Equations
• Mean:
• Standard deviation:
n
xx
n
ii
1
1
1
2
n
Xxn
ii
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Page 7Reid & Sanders, Operations Management© Wiley 2002
Impact of Standard Deviation
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Skewed Distributions (One Form of Non-Normal Distribution)
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Page 9Reid & Sanders, Operations Management© Wiley 2002
SPC Methods
• Control charts– Use statistical limits to identify when a
sample of data falls within a normal range of variation
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Page 10Reid & Sanders, Operations Management© Wiley 2002
Setting Limits RequiresBalancing Risks
• Control limits are based on a willingness to think something’s wrong, when it’s actually not (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error)
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Page 11Reid & Sanders, Operations Management© Wiley 2002
Types of Data
• Variable level data:– Can be measured using a continuous scale– Examples: length, weight, time, &
temperature
• Attribute level data:– Can only be described by discrete
characteristics– Example: defective & not defective
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Page 12Reid & Sanders, Operations Management© Wiley 2002
Control Charts for Variable Data
• Mean (x-bar) charts– Tracks the central tendency (the average
value observed) over time
• Range (R) charts:– Tracks the spread of the distribution over
time (estimates the observed variation)
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Page 13Reid & Sanders, Operations Management© Wiley 2002
x-Bar Computations
xx
xx
x
n
zxLCL
zxUCL
nn
xxxx
...21
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Example
• Assume the standard deviation of the process is given as 1.13 ounces• Management wants a 3-sigma chart (only 0.26% chance of alpha error)• Observed values shown in the table are in ounces
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample means 15.875 15.975 15.9
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Page 15Reid & Sanders, Operations Management© Wiley 2002
Computations
• Center line (x-double bar):
• Control limits:
92.153
9.15975.15875.15
x
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2nd Method Using R-bar
RAxLCL
RAxUCL
n
RRRR
x
x
n
2
2
21 ...
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Page 17Reid & Sanders, Operations Management© Wiley 2002
Control Chart Factors
Factor for x-ChartA2 D3 D4
2 1.88 0.00 3.273 1.02 0.00 2.574 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.82
10 0.31 0.22 1.7811 0.29 0.26 1.7412 0.27 0.28 1.7213 0.25 0.31 1.6914 0.24 0.33 1.6715 0.22 0.35 1.65
Factors for R-ChartSample Size (n )
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Page 18Reid & Sanders, Operations Management© Wiley 2002
Example
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample means 15.875 15.975 15.9
Sample ranges 0.2 0.3 0.2
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Page 19Reid & Sanders, Operations Management© Wiley 2002
Computations
22.1433.273.092.15
62.1733.273.092.15
33.23
2.03.02.0
2
2
RAxLCL
RAxUCL
R
x
x
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Page 20Reid & Sanders, Operations Management© Wiley 2002
Example x-bar Chart
X-bar Chart
12
13
14
15
16
17
18
1 2 3 4 5 6 7 8 9 10
Time
Ou
nc
es
UCL
LCL
CL
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R-chart Computations(Use D3 & D4 Factors: Table 6-1)
0033.2
71.628.233.2
33.23
2.03.02.0
3
4
DRLCL
DRUCL
R
R
R
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Example R-chart
R Chart
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10 11
Time
Ou
nc
es
UCL
LCL
CL
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Using x-bar & R-charts
• Use together• Reveal different
problems
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Control Charts for Attribute Data
• p-Charts:– Track the proportion defective in a sample
• c-Charts:– Track the average number of defects per
unit of output
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Page 25Reid & Sanders, Operations Management© Wiley 2002
Process Capability
• A measure of the ability of a process to meet preset design specifications:– Determines whether the process can do what we
are asking it to do
• Design specifications (a/k/a tolerance limits):– Preset by design engineers to define the
acceptable range of individual product characteristics (e.g.: physical dimensions, elapsed time, etc.)
– Based upon customer expectations & how the product works (not statistics!)
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Measuring Process Capability
Compare the width of design specifications & observed process output
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Capability Indexes
• Centered Process (Cp):
• Any Process (Cpk):
6 widthprocess
ion widthspecificat LSLUSLC p
3
;3
minLSLUSL
C pk
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Page 28Reid & Sanders, Operations Management© Wiley 2002
Example
• Design specifications call for a target value of 16.0 +/-0.2 microns (USL = 16.2 & LSL = 15.8)
• Observed process output has a mean of 15.9 and a standard deviation of 0.1 microns
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Page 29Reid & Sanders, Operations Management© Wiley 2002
Computations
• Cp:
• Cpk:
66.06.0
4.0
1.06
8.152.16
6
LSLUSL
C p
33.033.0or 1min3.0
1.0or
3.0
3.0min
1.03
8.159.15or
1.03
9.152.16min
3or
3min
LSLUSLC pk
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Three Sigma Capability
• Until now, we assumed process output should be modeled as +/- 3 standard deviations
• By doing so, we ignore the 0.26% of output that falls outside +/- 3 sigma range
• The result: a 3-sigma capable process produces 2600 defects for every million units produced
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Page 31Reid & Sanders, Operations Management© Wiley 2002
Six Sigma Capability
• Six sigma capability assumes the process is capable of producing output where +/- 6 standard deviations fall within the design specifications (even when the mean output drifts up to 1.5 standard deviations off target)
• The result: only 3.4 defects for every million produced
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3-Sigma versus 6-Sigma
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The End
Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions 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 contained herein.