TM 720 Lecture 07: Cont. Variable Charts, ARL & OC
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Transcript TM 720 Lecture 07: Cont. Variable Charts, ARL & OC
ENGM 720 - Lecture 08
P, NP, C, & U Control Charts
7/7/2015
ENGM 720: Statistical Process Control
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Outline
Assignment
Discrete
Distributions and Probability of Outcomes
• Examples of discrete distributions
Hypothesis
Testing to Control Charts
P- & NP-Charts
C- & U-Charts
Summary of Control Chart Options
• Using the Control Chart Decision Chart
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Assignment:
Reading:
•
•
Chapter 6
•
Finish reading
Chapter 7
•
•
•
Sections 7.1 and 7.2 through p.313
Sections 7.3 through p.325
Sections 7.3.2 and 7.5
Assignments:
•
•
•
Obtain the Control Chart Factors table from Materials Page
Access Excel Template for X-bar, R, & S Control Charts:
•
•
Download Assignment 5 for practice
Use the data on the HW5 Excel sheet to do the charting, verify the
control limits by hand calculations
Access Excel Template for P, NP, C, & U Control Charts
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Process for Statistical Control
Of Quality
Removing
special causes
of variation
Statistical Quality Control and Improvement
Improving Process Capability and Performance
• Hypothesis
Tests
• Ishikawa’s
Continually Improve the System
Characterize Stable Process Capability
Tools
Managing the
process with
control charts
Head Off Shifts in Location, Spread
• Process
Improvement
• Process
Stabilization
• Confidence in
Time
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
“When to Act”
Center the Process
LSL
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0
USL
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Review
Shewhart Control charts
•
•
•
Are like a sideways hypothesis test (2-sided!) from a
Normal distribution
• UCL is like the right / upper critical region
• CL is like the central location
• LCL is like the left / lower critical region
When working with continuous variables, we use two
charts:
• X-bar for testing for change in location
• R or s-chart for testing for change in spread
We check the charts using 4 Western Electric rules
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Continuous & Discrete Distributions
Continuous
• Probability of a range of
Discrete
• Probability of a range of
outcomes is area under
PDF (integration)
outcomes is area under
PDF (sum of discrete
outcomes)
35.0
2.5
30.4
(-3)
34.8
32.6
(-)
(-2)
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35.0
2.5
37
()
39.2
(+)
43.6
41.4
(+3)
(+2)
30
32
34
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36
()
38
40
6
42
Discrete Distribution Example
Sum
of two six-sided dice:
• Outcomes range from 2 to 12.
• Count the possible ways to obtain each individual sum - forms a
histogram
• What is the most frequently occurring sum that you could roll?
• Most likely outcome is a sum of 7 (there are 6 ways to obtain it)
• What is the probability of obtaining the most likely sum in a
single roll of the dice?
• 6 36 = .167
• What is the probability of obtaining a sum greater than 2 and
less than 11?
• 32 36 = .889
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Continuous & Attribute Variables
Continuous
Variables:
• Take on a continuum of values.
• Ex.: length, diameter, thickness
• Modeled by the Normal Distribution
Attribute
Variables:
• Take on discrete values
• Ex.:
present/absent, conforming/non-conforming
• Modeled by Binomial Distribution if classifying
inspection units into defectives
• (defective inspection unit can have multiple defects)
• Modeled by Poisson Distribution if counting defects
occurring within an inspection unit
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Discrete Variables Classes
Defectives
• The presence of a non-conformity ruins the
entire unit – the unit is defective
• Example – fuses with disconnects
Defects
• The presence of one or more nonconformities may lower the value of the unit,
but does NOT render the entire unit defective
• Example – paneling with scratches
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Binomial Distribution
Sequence of n trials
Outcome of each trial is “success” or “failure”
Probability of success = p
r.v. X - number of successes in n trials
X ~ Bin n, p
n x
n x
P X x p 1 p
x
So:
Mean: E X np
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where n
n!
x x! n x !
2
Variance: V X np 1 p
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Binomial Distribution Example
A lot of size 30 contains three defective fuses.
•
What is the probability that a sample of five fuses selected at
random contains exactly one defective fuse?
P[ X 1]
•
5 3
3
1
1 30 30
1
51
.328
(5)(.1)(.9)4
What is the probability that it contains one or more
defectives?
P[ X 1] 1 P[ X 0]
5 3
1
0 30
0
3
1
30
5 0
1 (1)(1)(.9)5
1 .5905
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.4095
11
Poisson Distribution
Let X be the number of times that a certain event occurs
per unit of length, area, volume, or time
X ~ Pois
So:
e x
P X x
x!
where x = 0, 1, 2, …
Mean: E X
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Variance: 2 V X
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Poisson Distribution Example
A sheet of 4’x8’ paneling (= 4608 in2) has 22 scratches.
•
•
What is the expected number of scratches if checking only
one square inch (randomly selected)?
22
λ1
.00477
4608
What is the probability of finding at least two scratches in 25
in2?
25
.119
λ25
λ1 25( λ1 ) 25(.00477)
i 1
P[ X 2] 1 P[ X 0] P[ X 1]
e .119 (.119) 0 e .119 (.119)1
.888(1) .888(.119)
1
1 (.888 .106)
1
1
1
0
!
1
!
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.007
Moving from Hypothesis Testing
to Control Charts
Attribute control charts are also like a sideways hypothesis test
• Detects a shift in the process
• Heads-off costly errors by detecting trends –
if constant control limits are used
2
2
2
0
2-Sided Hypothesis Test
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UCL
0
CL
2
Sideways Hypothesis
Test
LCL
Sample Number
Shewhart Control Chart
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P-Charts
Tracks proportion defective in
Can
a sample of insp. units
have a constant number of inspection units in the sample
Sample Control Limits:
• Approximate 3σ limits are
found from trial samples:
UCL p 3
p(1 p )
n
Standard Control Limits:
• Approximate 3σ limits
continue from standard:
UCL p 3
CL p
CL p
p(1 p )
LCL p 3
n
LCL p 3
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p(1 p)
n
p(1 p)
n
15
P-Charts (continued)
More
commonly has variable number of inspection units
Can’t
use run rules with variable control limits
Mean Sample Size Limits:
• Approximate 3σ limits are
found from sample mean:
UCL p 3
p(1 p )
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Variable Width Limits:
• Approximate 3σ limits vary
with individual sample size:
UCL p 3
n
p(1 p)
ni
CL p
CL p
LCL p 3
p(1 p )
LCL p 3
n
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p(1 p)
ni
16
NP-Charts
Tracks number
of defectives in a sample of insp. units
Must
have a constant number of inspection units in each sample
Use of run rules is allowed if LCL > 0 - adds power !
Sample Control Limits:
Standard Control Limits:
• Approximate 3σ limits are
found from trial samples:
• Approximate 3σ limits
continue from standard:
UCL np 3 np(1 p)
UCL np 3 np(1 p)
CL np
CL np
LCL np 3 np(1 p)
LCL np 3 np(1 p)
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C-Charts
Tracks number
of defects in a logical inspection unit
Must
have a constant size inspection unit containing the defects
Use of run rules is allowed if LCL > 0 - adds power !
Sample Control Limits:
• Approximate 3σ limits are
found from trial samples:
UCL c 3 c
CL c
LCL c 3 c
Standard Control Limits:
• Approximate 3σ limits
continue from standard:
UCL c 3 c
CL c
LCL c 3 c
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or 0 if LCL is negativ e
or 0 if LCL is negativ e
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U-Charts
Number of defects occurring in variably sized inspection
(Ex. Solder defects per 100 joints - 350 joints in board = 3.5 insp. units)
Can’t use run rules with variable control limits, watch clustering!
Mean Sample Size Limits:
• Approximate 3σ limits are
found from sample mean:
UCL u 3
u
Variable Width Limits:
• Approximate 3σ limits vary
with individual sample size:
UCL u 3
n
CL u
LCL u 3
n
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u
ni
CL u
u
unit
LCL u 3
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u
ni
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Steps for Trial Control Limits
Start with 20 to 25 samples
Use all data to calculate initial control limits
Plot each sample in time-order on chart.
Check for out of control sample points
•
•
If one (or more) found, then:
1. Investigate the process;
2. Remove the special cause; and
3. Remove the special cause point and recalculate
control limits.
If can’t find special cause - drop point & recalculate
anyway
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Summary of Control Charts
Use
of the control chart decision table.
Continuous
Variable Charts
• Smaller changes detected
faster
• Apply to attributes data as well
(by CLT)*
• Require smaller sample sizes
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Attribute
Charts
• Can cover several
defects with one chart
• Less costly inspection
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Control Chart Decision Table
Defective Units
(possibly with multiple defects)
Binomial Distribution
Is the size of
the inspection
sample fixed?
No, varies
Use p-Chart
Yes,
constant
Discrete
Attribute
What is the
inspection
basis?
Individual Defects
Poisson Distribution
Is the size of
the
inspection
unit fixed?
Kind of
inspection
variable?
Use np-Chart
Yes,
constant
Use c-Chart
No, varies
Continuous
Variable
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Which spread
method
preferred?
Use u-Chart
Range
Use X-bar and
R-Chart
Standard Deviation
ENGM 720: Statistical Process Control
Use X-bar and
S-Chart
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Control Chart Sensitizing Rules
Western Electric Rules:
1.
One point plots outside the three-sigma limits;
2.
Eight consecutive points plot on one side of the center line
(run rule!);
3.
Two out of three consecutive points plot beyond two-sigma
warning limits on the same side of the center line (zone rule!);
3.
Four out of five consecutive points plot beyond one-sigma
warning limits on the same side of the center line (zone rule!).
If chart shows lack of control, investigate for special cause
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Attribute Chart Applications
Attribute control charts apply to “service”
applications, too.
• Number of incorrect invoices per customer
• Proportion of incorrect orders taken in a day
• Number of return service calls to resolve problem
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Questions & Issues
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