Transcript Introduction to Statistical Quality Control, 5th edition

IES 331 Quality Control
Chapter 6
Control Charts for Attributes
Week 7-8
July 19-28, 2005
Dr. Karndee Prichanont
IES331 1/2005
Attribute Data



If item does not conform to standard on
one or more of these characteristics, it is
classified as nonconforming

Conforming / Nonconforming units

Non-defective / Defecting units

Good / Bad

Pass / Fail
Nonconforming unit will contain at least
on nonconformity
Nonconformities / Defects

Each specific point at which a specification
is not satisfied

ex: scratch, chip, dirty spots, accident
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Dr. Karndee Prichanont
IES331 1/2005
6-2 Control Chart for
Fraction Nonconforming

Fraction of Nonconforming: Ratio of the
number of nonconforming items in a population to
the total number of items in that population

Fraction of nonconforming ~ Binomial Distribution
p
n
D
Probability that any unit will not conform to specifications
Random sample of n unit. Sample size
The number of units of products that are nonconforming
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Dr. Karndee Prichanont
IES331 1/2005
The Control Chart for
Fraction Nonconforming
Sample fraction nonconforming:
ratio of the number of
nonconforming units in the sample,
D, to the sample size n
Mean and Variances
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Dr. Karndee Prichanont
IES331 1/2005
The Control Chart for
Fraction Nonconforming
With specified standard value
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Dr. Karndee Prichanont
IES331 1/2005
The Control Chart for
Fraction Nonconforming
When p is not known, it
must be estimated
from collected data
Average of these individual
sample fractions
nonconforming
Fraction Nonconforming
control chart: No
Standard Given
“Trial Control Limit”
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Dr. Karndee Prichanont
Ex 1
(Exercise 6-1)
Also see Example 6-1
The data that follow give the
number of nonconforming
bearing and seal
assemblies in sample size
of 100
Construct a fraction
nonconforming control
chart for these data.
If any points plot out of
control, assume that
assignable causes can be
found and determine the
revised control limits
IES331 1/2005
Sample #
Number of
Noncomforming
Assemblies
1
7
2
4
3
1
4
3
5
6
6
8
7
10
8
5
9
2
10
7
11
6
12
15
13
0
14
9
15
5
16
1
17
4
18
5
19
7
20
12
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Dr. Karndee Prichanont
IES331 1/2005
The np Control Chart




Alternative to p Control Chart
Based on the number nonconforming rather than
the fraction nonconforming
If standard value p is not known, use the
estimator
Revisit Ex 1
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Dr. Karndee Prichanont
IES331 1/2005
Variable Sample Size
3 approaches to deal with variable sample size
1.
Variable-width control limits: to determine control
limits for each individual sample that are based on
specific sample size
2.
Control limits based on average sample size: to
obtain an approximate set of control limits (constant
control limits)
3.
The standardized control chart:

The points are plotted in standard deviation units

Center line at zero

Upper and lower control limits  +3 and - 3
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Dr. Karndee Prichanont
Ex 2
IES331 1/2005
(Exercise 6-3)
Also see Example 6-2
The following data
represent the results of
inspecting all units of a
personal computer
produced for the last 10
days.
Does the process appear to
be in control?
Day
Units
Inspe
cted
Nonconforming
units
Fraction
Nonconfo
rming
1
80
4
0.050
2
110
7
0.064
3
90
5
0.056
4
75
8
0.107
5
130
6
0.046
6
120
6
0.050
7
70
4
0.057
8
125
5
0.040
9
105
8
0.076
10
95
7
0.074
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Dr. Karndee Prichanont
IES331 1/2005
The Operating-Characteristic Function

OC curve provides a measure
of the sensitivity of the control
chart


Ability to detect the shift in the
process fraction nonconforming
from the nominal value to
some other value
Probability of incorrectly
accepting the hypothesis of
statistical control (i.e., type II
or β-error)
  P{ pˆ  UCL | p}  P{ pˆ  LCL | p}
 P{D  nUCL | p}  P{D  nLCL | p}
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Dr. Karndee Prichanont
IES331 1/2005
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Dr. Karndee Prichanont
IES331 1/2005
Average Run Length (ARL)



ARL for any Shewhart
control chart
If the process in control,
ARL0
If the process is out of
control, ARL1
ARL 
1
P(sample point plots out of control)
ARL0 
1

1
ARL1 
1 
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Dr. Karndee Prichanont
IES331 1/2005
6-3 Control Charts for Nonconformities
(Defects)


We can develop control
charts for

Total number of
nonconformities in a
unit, or

Average number of
nonconformities per unit
Defects or
nonconformities ~ Poisson
Distribution
c chart:
same sample size
Where x is the number of
nonconformities and C > 0
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Dr. Karndee Prichanont
Ex 3
(Exercise 6-41)
Also see Example 6-3
The data represent the number
of nonconformities per 1000
meters in telephone cable.
From analysis of these data,
would you conclude that the
process is in control?
Sample Number
IES331 1/2005
Number of
Nonconformities
1
1
2
1
3
3
4
7
5
8
6
10
7
5
8
13
9
0
10
19
11
24
12
6
13
9
14
11
15
15
16
8
17
3
18
6
19
7
20
4
21
9
22
20
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Dr. Karndee Prichanont
IES331 1/2005
Choice of Sample Size:
u chart


It may requires several inspection
units in the sample
Average number of
nonconformities per
inspection unit
u chart: setting up the control
chart based on the average
number of nonconformities per
inspection unit
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Dr. Karndee Prichanont
Ex 4
(Exercise 6-44)
Sample Number
Number of
Nonconformities
Also see Example 6-4
1
1
2
3
An automobile manufacturer
wishes to control the
number of
nonconformities in a
subassembly area
producing manual
transmissions.
3
2
4
1
5
0
6
2
7
1
8
5
The inspection unit is defined
as four transmissions, and
data from 16 samples
(each of size 4) are
shown here.
9
2
10
1
11
0
12
2
13
1
14
1
15
2
16
3
Set up a control chart for
nonconformities per unit
IES331 1/2005
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Dr. Karndee Prichanont
IES331 1/2005
Control charts for nonconformities with
variable sample size

c chart will be difficult to interpret –
varying center line and control limits

Alternative approaches are:
1.
u chart (nonconformities per unit)

Constant center line

Control limit will vary inversely with
the n1/2
2.
Use control limits based on an average
sample size
3.
Use a standardized control chart
(preferred)
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Dr. Karndee Prichanont
IES331 1/2005
Demerit Systems for Defects
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Dr. Karndee Prichanont
IES331 1/2005
The Operating Characteristic Function

OC curves for c and u charts
can be obtained from the
Poisson distribution
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Dr. Karndee Prichanont
IES331 1/2005
Action taken to improve a process
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