Transcript Operations

CHAPTER
Statistical Quality Control
Reid & Sanders, Operations Management
© Wiley 2002
6
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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© Wiley 2002
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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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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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The Normal Distribution
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Equations
• Mean:
n
x
x
i
i 1
n
• Standard deviation:
 x  X 
n

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i 1
2
i
n 1
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Impact of Standard Deviation
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Skewed Distributions
(One Form of Non-Normal Distribution)
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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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Setting Limits Requires
Balancing 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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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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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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x-Bar Computations
x1  x 2  ... x n
x
n
x 

n
UCLx  x  z x
LCLx  x  z x
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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
15.975
15.9
Sample means 15.875
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Computations
• Center line (x-double bar):
15.875  15.975  15.9
x
 15.92
3
• Control limits:
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2nd Method Using R-bar
R1  R2  ... Rn
R
n
UCLx  x  A2 R
LCLx  x  A2 R
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Control Chart Factors
S a m p l e S i z e (n )
F a c to r fo r x -C h a r t
F a c to r s fo r R -C h a r t
A2
D3
D4
2
1.88
0.00
3.27
3
1.02
0.00
2.57
4
0.73
0.00
2.28
5
0.58
0.00
2.11
6
0.48
0.00
2.00
7
0.42
0.08
1.92
8
0.37
0.14
1.86
9
0.34
0.18
1.82
10
0.31
0.22
1.78
11
0.29
0.26
1.74
12
0.27
0.28
1.72
13
0.25
0.31
1.69
14
0.24
0.33
1.67
15
0.22
0.35
1.65
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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
15.975
15.9
0.3
0.2
Sample means 15.875
Sample ranges
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0.2
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Computations
0.2  0.3  0.2
R
 2.33
3
UCLx  x  A2 R  15.92  0.732.33  17.62
LCLx  x  A2 R  15.92  0.732.33  14.22
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Example x-bar Chart
X-bar Chart
18
UCL
17
Ounces
16
CL
15
14
LCL
13
12
1
2
3
4
5
6
7
8
9
10
Time
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R-chart Computations
(Use D3 & D4 Factors: Table 6-1)
0.2  0.3  0.2
R
 2.33
3
UCLR  RD4  2.332.28  6.71
LCLR  RD3  2.330   0
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Example R-chart
R Chart
8
7
UCL
6
Ounces
5
4
3
CL
2
1
0
LCL
1
2
3
4
5
6
7
8
9
10
11
Time
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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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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):
specificat ion width USL  LSL
Cp 

process width
6
• Any Process (Cpk):
C pk
 USL     LSL 
 min 
;

3 
 3
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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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Computations
• C p:
• Cpk:
USL  LSL 16.2  15.8 0.4
Cp 


 0.66
6
60.1
0.6
 USL     LSL 
C pk  min 
or

3 
 3
 16.2  15.9 15.9  15.8 

 min 
or
30.1 
 30.1
 0.3 0.1 
 min 
or
  min 1 or 0.33  0.33
 0.3 0.3 
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© Wiley 2002
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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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© Wiley 2002
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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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© Wiley 2002
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3-Sigma versus 6-Sigma
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© Wiley 2002
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The End
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