Download 7.1 Introduction

Learning Objectives
Chapter 7
Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.
Copyright (c) 2009 John Wiley & Sons, Inc.
3
7.1 Introduction
Many quality characteristics are not measured on a continuous scale
or even a quantitative scale. In such cases, one may judge each unit of
product as either conforming or nonconforming on the basis of
whether or not it possesses certain attributes or we may count the
number of nonconformities (defects) appearing on the unit of
product. Control charts for such quality characteristics are called
attributes control charts.
1
7.2 The Control Chart for Fraction nonconforming
The population fraction nonconforming (FNC) is defined as the
ratio of the number of nonconforming items in a population to the
total number of items in that population. Similarly, the sample
fraction nonconforming is defined as the ratio of the number of
nonconforming items in a sample to the number of items in the
corresponding sample. The item may have several quality
characteristics that are examined simultaneously by the inspector. If
the item does not conform to standard on one or more of these
characteristics, it is classified as nonconforming.
The statistical
principles underlying the control chart for FNC are based on the
binomial distribution. We assume that the process is operating in a
stable manner, such that the probability that any unit will not
conform to specifications is p and successive units produced are
independent. Then each unit produced is a realization of Bernoulli
random variable with parameter p . Suppose that each subgroup is of
the same size n , and D denotes the random variable that counts the
number of nonconforming items in a subgroup. Then D can be
modeled as a binomial random variable with parameters n and p .
n
P ( D = x) =   p x (1  p ) n  x ; x = 0,1,2, , n
 x
That is
Then
 D = np,  D2 = np(1  p)
The sample fraction nonconforming corresponding to population
nonconforming is obtained as:
pˆ =
D
.
n
It can be shown that
 D  np
E ( pˆ ) =  pˆ = E   =
=p
n n
and
np(1  p) p (1  p )
1
Var ( D) =
=
.
2
n
n2
n
So the mean and standard deviation of p̂ are respectively
Var ( pˆ ) =  p2ˆ =
 pˆ = p, and  pˆ =
2
p (1  p )
.
n
(1)
7.2.1 Development and Operation of the Control Chart
Let w be a sample statistic that measures some quality characteristic
of interest, and suppose,
E ( w) =  w
V(w) =  w2 ,  SD( w) =  w
and
Then the UCL, center line, and LCL become:
UCL =  w  L w
(2)
CL =  w
LCL =  w  L w ,
where L (usually 3) is the ``distance'' of the control limits from the
center line, expressed in standard deviation units. This is called
Shewhart (Dr. Walter A. Shewhart) Control chart.
Fraction Nonconforming Control Chart: Standard Given
UCL = p  3
p(1  p)
n
(3)
CL = p
LCL = p  3
p(1  p )
n
The actual operation of this chart would consist of taking subsequent
samples of n units. Computing the sample fraction nonconforming
p̂i and plotting the statistic p̂i versus its subgroup number i on the
chart. As long as p̂ remains within the control limits and the
sequence of plotted points does not exhibit any systematic
nonrandom pattern, we may conclude that the process is in control at
the level p . If a point plots outside of the control limits, or if a
nonrandom pattern in the plotted points is observed, we may
conclude that the process fraction nonconforming most likely shifted
to a new level and the process is out of control.
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Fraction Nonconforming (FNC) Control Chart: No Standard Given
UCL = p  3
p (1  p )
n
(4)
CL = p
LCL = p  3
p (1  p )
n
The control limits in (4) should be considered as trial control limits.
Estimation of p
When the process fraction nonconforming p is not known, then it
must be estimated from observed data. The usual procedure is to
select m preliminary samples (20 to 30), each of size n (5 to 10).
Case 1: Suppose there are Di nonconforming items in the sample i ,
then we compute the fraction nonconforming for the i th sample as
pˆ i =
Di
,
n
i = 1,2, , m.
And the average of these individual sample FNC is
m
m
D  pˆ
i
i
p=
i =1
i =1
=
nm
m
Case 2: When the subgroup sizes n1 , n2 ,..., nm are not all equal, then the
p can be calculated as
D  D2 ,..., Dm
p= 1
n1  n 2 ,..., n m
D 
D  D 
=  1    2   ...   m 
N   N 
 N 
n 
n 
n 
=  1  pˆ 1   2  pˆ 2  ...   m  pˆ m
N
N
N 
m
= wi pˆ i ,
i =1
n
where wi = i and n1  n2  ...  nm = N and p is a weighted average of
N
the subgroup statistics.
4
(5)
Example 7.1, page 292
From Table 7.1, we calculate the following preliminary control limits.
m
D
i
p=
i =1
=
nm
347
= 0.2313
30  50
The upper control limit, center line and the lower control limits are
p (1  p )
= 0.4102
n
CL = p = 0.2313
UCL = p  3
LCL = p  3
(6)
p (1  p )
= 0.0524
n
Figure 7.1 Initial phase I fraction nonconforming control chart
The control chart has shown in Figure 7.1. We observed that samples
15 and 23 plot above the upper control limit. Therefore, the process is
not in control. Since the samples 15 and 23 are out of control limits
they are eliminated and the revised control limits are as follows:
m
D
i
p=
i =1
=
nm
301
= 0.2150
28  50
p (1  p )
= 0.3893
n
CL = p = 0.2150
UCL
LCL
= p  3
= p  3
p (1  p )
= 0.0407
n
5
(7)
The revised control chart has shown in Figure 7.2. We observed that
samples 15 and 23 plot above the upper control limit, even they have
been excluded from the calculation of the control limits. In the
revised control limits, the sample number 21 is out of the limit.
However, there is no assignable causes related to that sample. So we
conclude that the process is in control at level p = 0.2150 and the
revised control limits should be used for monitoring current
production.
Figure 7.2 Revised control limits for the data in Table 7.1 (page 293)
During the next three shifts following the machine adjustments and
the introduction of the control chart, an additional 24 samples of size
n = 50 observations each are collected and provided them in Table 7.2,
page 295. The sample fraction nonconforming are plotted on the
control chart in Fig 7.3.
Figure 7.3: Continution of fraction nonconforming control chart.
6
From Fig 7.3, we see that the process is now operating at a new level
which is lower than the present level p = 0.2150 . Now, we are
interested for the following hypothesis
H 0 : p1 = p2
H1 : p1 > p2
We have pˆ 1 = 0.2150 , and
54
D
i
pˆ 2 =
i = 31
50  24
=
133
= 0.1108
50  24
The approximate Z-test (see more on page 296) is
Z0 =
Z0 =
pˆ 1  pˆ 2
pˆ (1  pˆ )1/n1  1/n2 
0.2150  0.1108
= 7.10
(0.1669)(0.8331)1/1400  1/1200 
Since Z 0 = 7.10 > 1.645 , we do reject the null hypothesis and conclude
that there has been a significant decrease in the process fallout. The
revised control limits based on the last 24 samples (numbers 31-54)
are
p (1  p )
= 0.2440
n
CL = p = 0.1108
UCL = p  3
LCL = p  3
p (1  p )
= 0.0224 = 0
n
The new revised control chart has shown in Figure 7.4.
Figure 7.4: New control limits on FNC control chart, Example 7.1
7
(8)
The new control limits will have only upper control limit. All points
are inside the revised control limits. Therefore the process is in
control at this new level.
The continued operation of this control chart for the next five shifts
(data in Table 7.3, page 298) is shown in Fig 7.5.
Figure 7.5: Completed fraction nonconforming CC for Example 7.1
The process in control. However, the fraction nonconforming is still
high. You need Experimental Design to reduce the process fraction
nonconforming. See more on page 297.
Design of the Fraction Nonconforming Control Chart (FNCC)
The FNCC has three parameters: the sample size, the frequency of
sampling and the width of the control limits.
Various rules have been suggested for the choice of sample size n .
 When p is small, larger subgroup sizes are necessary, and for
larger p , smaller subgroups are necessary.
 We can choose the sample size n so that the probability of finding
at least one nonconforming unit per sample is at least  . If D denotes
the number of NCF items in the sample, then we want to find n such
that
p{D  1}   .
8
Using Poisson approximation to binomial, we find that
n=
ln(1   )
.
p
(9)
Example, if P( D  1)  0.95 , & p = 0.01 , then sample size should be 300.
Example, if P( D  1)  0.99 , & p = 0.01 , then sample size should be 461.
 Duncan (1986) suggested that the sample size should be large
enough such that we have approximately 50% chance of detecting a
process shift of some specified amount. If  is the magnitude of the
process shift, then
2
L
n =   p(1  p)
 
(10)
For example, p = 0.01 , we want to detect a shift when p = 0.05 . Then
 = 0.05  0.01 = 0.04 . And the sample size for 3-sigma control limit
would be
2
 3 
n=
 0.01(1  0.01) = 56
 0.04 
For example, p = 0.01 , we want to detect a shift when p = 0.02 . Then
 = 0.02  0.01 = 0.01 . And the sample size for 3-sigma control limit
would be
2
 3 
n=
 0.01(1  0.01) = 891
 0.01 
For a smaller shift, you need a bigger sample size.
 What is the smallest sample size that would give a positive lower
limit?
LCL  p  L
n>
p(1  p )
>0
n
(1  p ) 2
L .
p
For example, if p = 0.05 , and 3-sigma limits are used, the sample size
must be n  171 .
For example, if p = 0.01 , and 3-sigma limits are used, the sample size
must be n  892 .
9
The np Control Chart
limits are as follows:
The number nonconforming or np control
UCL = np  3 np(1  p )
(11)
CL = np
LCL = np  3 np (1  p)
If p is unknown, use p to estimate p .
Example 7.2, page 300.
Consider the data in Table 7.1 of example 7.1. The control limits for
np chart would be
UCL = np  3 np (1  p ) = 20.51 = 20
CL = np = 11.57 = 12
(12)
LCL = np  3 np (1  p ) = 2.62 = 2
From Table 7.1, the sample number 15 and 23 are out of control.
7.2.2 Variable Sample Size
There are three approaches to constructing and operating a control
chart.
1. Variable-width Control Limits
The control limits for the i th sample are
p (1  p )
ni
UCL = p  3
(13)
CL = p
p (1  p )
ni
LCL = p  3
m
Replace p with p , where, p =
D
i
i =1
m
n
.
i
i =1
10
Consider data in Table 7.4, page 302 we calculate
25
D
i
p=
i =1
25
n
=
234
= 0.096
2450
i
i =1
then the center line is 0.096. The control limits are
UCL = 0.096  3
0.096  0.904
0.884
= 0.096 
ni
ni
(14)
CL = 0.096
LCL = 0.096  3
0.096  0.904
0.884
= 0.096 
ni
ni
The control chart for fraction nonconforming with variable sample
size is provided in Fig 7.6.
Figure 7.6: CC for FNC with variable sample size
11
2. Control limits based on an average sample size
m
n
i
n=
i =1
m
Then the approximate control limits are
p (1  p)
n
UCL = p  3
(15)
CL = p
p (1  p )
n
LCL = p  3
Consider data in Table 7.4, page 302 we calculate
25
D
i
n=
i =1
25
n
=
2450
= 98
25
i
i =1
Then the control limits are
UCL = 0.096  3
0.096  0.904
= 0.185
98
(16)
CL = 0.096
LCL = 0.096  3
0.096  0.904
= 0.007
98
The resulting control chart is shown in Fig 7.8.
Figure 7.8: CC for FNC based on the average sample size
12
3. The standardized control chart
The standardized control chart has center line at 0, upper and lower
limits of +3 and -3 respectively. The variable plotted on the chart is
Zi =
pˆ i  p
p(1  p)
ni
where p (or p , if standard is not given) is the process fraction
nonconforming in the in-control state. The standardized control chart
for fraction nonconforming is presented in Fig 7.9 for the data in
Table 7.5, page 305.
Figure 7.9: Standardized CC for FNC
7.2.3 Application in Transactional and Service Business: pages 304306
7.2.4 The Operating Characteristic (OC) Function and Average Run
Length (ARL)
The OC function of the fraction nonconforming control chart is a
graphical display of the probability of incorrectly accepting the
hypothesis of statistical control (i.e. type II error or  -error) against
the process fraction nonconforming. The OC curve provides a
measure of the sensitivity of the control chart, that is, its ability to
detect a shift in the process fraction nonconforming from the nominal
value p to some other value.
13
The probability of type II error is
 = P{LCL < pˆ < UCL | p}
D
< UCL | p}
n
= P{D < n  UCL | p}  P{D  n  LCL | p}
 P{LCL <
Since D is the binomial random variable with parameters n and p ,
the  error defined in (17) can be obtained from binomial table. Table
7.6, page 307 illustrates the calculations of a OC curve for n = 50 ,
LCL = 0.0303 and UCL = 0.3697 . The corresponding OC curve has shown
in Fig 7.11.
14
(17)
The average run length (ARL) for any Shewhart control chart is
defined as
ARL =
1
Sample point plots out of control
(18)
If the process is in control, ARL0 is defined as
ARL0 =
1
(19)

For example, if the process is in control with p = 0.20 , the probability
of a point plotting in control is 0.9973 (from Table 7.6). Then
ARL0 =
1
= 370
1  0.9973
Then, even the process is in control, we will experience a false alarm
out of control signal about every 370 samples.
However, if the process is out of control, then
ARL1 =
1
1 
(20)
For example, if the process shift to p = 0.30 , Table 7.6 indicates that
 = 0.8594 . Then
ARL1 =
1
=7
1  0.8594
and it will take about 7 samples, on average, to detect this shift with a
point outside of the control limits.
Exercise 7.7, page 336.
15
Exercise 7.10, page 336.
Exercise 7.18, page 337.
7.3 Control Charts for Nonconformities (Defects)
A nonconforming item is a unit of product that does not satisfy one
or more of the specifications for that product. Each specific point at
which a specification is not satisfied results in a defect or
nonconformity. Consequently, a nonconforming item will contain at
least one nonconformity. It is possible to construct control chart
based on the total number of nonconformities in a unit or the average
number of nonconformities per unit.
7.3.1 Procedures with Constant Sample Size
Consider the
occurences of nonconformities in an inspection unit of product. The
inspection unit is simply an entry for which it is convenient to keep
records. Suppose that defects or nonconformities occur in the
inspection unit according to the Poisson distribution. That is,
p ( x) =
ecc x
,
x!
where x is the number of nonconformities and c is the parameter of
Poisson distribution. The control chart for nonconformities with 3sigma control limits are defined as follows:
Control chart for Nonconformities: Standard Given (c-chart)
UCL = c  3 c
(21)
CL = c
LCL = c  3 c
If the LCL is a negative value, consider as LCL=0.
16
Control chart for Nonconformities: No Standard Given
UCL = c  3 c
(22)
CL = c
LCL = c  3 c
When no standard is given, the control limits in equation (22) should
be regarded as trial control limits.
Example 7.3, page 310
Table 7.7 (page 310) represents the number of nonconformities
observed in 26 successive samples of 100 printed circuit boards. For
reasons of convenience, the inspection unit is defines as 100 boards.
Construct control limits for the c-Chart. Since 26 samples contain
516 total nonconformities, we have
c=
516
= 19.85.
26
The trial control limits are
UCL = c  3 c = 33.22
(23)
CL = c = 19.85
LCL = c  3 c = 6.48
The control chart based on limits in (23) is shown in Fig 7.12.
Figure 7.12: CC for NConformities for Example 7.3
17
From Fig 7.12, we observe that Samples 6 and 20 plot outside the
control limits. Then exclude these two samples and revised control
limits which are obtained as follows:
c=
472
= 19.67.
24
The revised control limits are
UCL = c  3 c = 32.97
(24)
CL = c = 19.67
LCL = c  3 c = 6.37
The above control limits become the standard values against which
production in the next period can be compared.
Twenty new
samples are collected and given them in Table 7.8, page 311. Using
revised control limits in (24), these points are plotted in Fig 7.13. It is
evident that this process in statistical control.
Figure 7.13: Continution of the CC for NConformities, Example 7.3
Exercise 7.36, 338.
18
Control chart for average number of nonconformities per unit (uchart)
To account for the variable numbers of inspection units, c charts are
replaced by the u charts. The u chart monitors nonconformities per
inspection unit, when the number of inspection units are vary from
sample to sample. If a total of x nonconformities are found in the
subgroup of size n inspection units, then the average number of
nonconformities per unit can be estimated as
u=
x
,
n
where x is a Poisson random variable. The control limits for u chart
are:
UCL = u  3
u
n
(25)
CL = u
LCL = u  3
u
,
n
where
m
u
u=
i =1
m
i
.
Example 7.4, page 315 Data are presented in Table 7.10 (page 316).
20
u
u=
i
i =1
20
=
1.48
= 0.0740
20
The upper control limit, center line and lower control limits are
UCL = u  3
u
= 0.1894
n
(26)
CL = u = 1.93
LCL = u  3
u
= 0.0414  0
n
The control chart for nonconformities is presented in Fig 7.16. The
preliminary data do not exhibit lack of statistical control. Therefore,
the trial limits in (26) would be adopted for current control purposes.
19
Figure 7.16: The CC for NConformities per unit for Example 7.4
Exercise 7.38, Page 338.
6-3.2 Procedures with Variable Sample Size
There are three
approaches to constructing and operating a control chart.
1. Variable-width Control Limits: The control limits for the i th
sample are
UCL = u  3
u
ni
(27)
CL = u
LCL = u  3
u
ni
Consider data in Table 7.11, page 319 we calculate
u=
153
= 1.42
107.5
The upper control limit, center line and lower control limits are
UCL = 1.42  3
1.42
3.575
= 1.42 
ni
ni
CL = 1.42
LCL = 1.42  3
1.42
3.575
 1.42 
ni
ni
20
The control chart for fraction nonconforming with variable sample
size is provided in Fig 7.17.
Figure 7.17: CC for Example 7.5
2. Control limits based on an average sample size
The approximate control limits are
UCL = u  3
u
n
(28)
CL = u
LCL = u  3
u
,
n
where
m
n
i
n=
i =1
m
.
3. The standardized control chart
The standardized control chart has center line at 0, upper and lower
limits of +3 and -3 respectively. The variable plotted on the chart is
Zi =
uˆi  u
u
ni
(29)
The standardized control chart for nonconformities per unit is
presented in Fig 7.18 for the data in Table 7.11.
21
Figure 7.18: Standardized CC for NConformities per unit, Example
7.5
22
7.3.4 The Operating -Characteristic (OC) Function
The OC curves for both the c and u charts can be obtained from the
Poisson distribution. For c chart, the OC curve plots the probability
of type II error against the true mean number of defects c . The
expression for  is
 = P{LCL < x < UCL | c}
(30)
= P{x < UCL | c}  P{x  LCL | c}
where x is a Poisson random variable with parameter c . Table 7.13,
page 323 illustrates the calculations for a OC curve for example 7.3,
LCL = 6.48 and UCL = 33.22 . The corresponding  is
(31)
 = P{x  33 | c}  P{x  6 | c}
The OC curve has shown in Fig 7.19, page 324.
Figure 7.19: OC Curve of a c chart with LCL=6.48 and UCL=33.22
The OC curve for u chart is
 = P{x < UCL | u}  P{x  LCL | u}
 = P{c < n  UCL | u}  P{c  n  LCL | u}
Exercise 7.56, Page 340.
23
(32)
7.3.5 Dealing with Low Defect Levels
When defect levels (or count rates) in a process become very low (less
than 1000 per million), there will be very long periods of time
between the occurrence of nonconforming unit. In these situations
many samples will have zero defects and therefore, conventional c
and u chart become ineffective. See more in Example 7.6, page 324.
7.4 Choice between attributes and variables control
charts
Read page 326.
Example 7.7, page 327, demonstrates the economic advantage of
variable control chart.
Example 7.8, page 328, demonstrates a misapplication of x and R
charts.
7.5 Guidelines for implementing control charts (pages 330-334)
1. Determining which process characteristics to control
2. Determining where the charts should be implemented in the
process
3. Choosing the proper type of control charts
4. Taking actions to improve processes as the result of SPC/ control
chart analysis.
5. Selecting data-collection systems and computer software
24