Control Charts For Attributes 2
Control Charts For Attributes 2
Control Charts For Attributes 2
Session 2.1 :
Control Charts for Nonconformity
A nonconformity is a quality characteristic that does not meet specifications A nonconforming item has one or more nonconformities If there are 3 scratches on an item then number of nonconformities is counted as 3 Size of the sample is called as area of opportunity
Area of opportunity is the area that you are looking for nonconformities Examples are 100 m2 of fabric, 10 TV sets, 1 roll of paper This area must be chosen wide enough that there exist a number of nonconformities If the average number of nonconformities for a TV set is 0.08, then sample sizes of 50 would make sense (rather than 10)
C Chart
u Chart
Example :
In a process of manufacturing the circuit boards, the number of nonconformities was observed in 26 samples, each sample for reason of convenience was 100 boards. Construct a control chart to control the process.
Sample No. 1 2 3 4 5 6 7 8 9 10 No. of nonconformities 21 24 16 12 15 5 28 20 31 25 Sample No. 14 15 16 17 18 19 20 21 22 23 No. of nonconformities 19 10 17 13 22 18 39 30 24 16
11
12 13
20
24 16
24
25 26
19
17 15
We noticed that two points plot outside the control limits. 6 and 20
Investigation of sample 6 revealed that new inspector had examined the boards in this sample and he didnt recognize several of the types of nonconformities that could have been present. Furthermore, the unusually large number of nonconformities in sample 20 resulted from a temperature control problem in the wave soldering machine.
Therefore, it seems reasonable to exclude these two samples and revise the trial control limits.
These becomes the standard values against which production in the next period can be compared.
No lack of control is indicated, the process is in control on this level. However, The number of nonconformities per board is still unacceptably high. Management action is necessary to improve the process.
Workshop :
The number of nonconformities found on final inspection of a cassette deck is shown here. Can you conclude that the process is in statistical control? What center line and control limits would you recommend for controlling future production? What are the center line and control limits for a control chart for monitoring future production based on the total number of defects in a sample of 4 cassette decks?
Deck No 1 2 3 4 5 6 7 8 no of Nonconformities 0 1 1 0 2 1 1 3 Deck No 10 11 12 13 14 15 16 17 no of Nonconformities 1 0 3 2 5 1 2 1
18
There is no reason why the sample size must be restricted to one inspection unit. We would often prefer to use several inspection units in the sample, thereby increasing the area of opportunity for occurrence of nonconformities.
to illustrate this suppose that we were to specify a subgroup size of n = 2.5 inspection units.
to construct a chart once a new sample size has been selected. You can use a control chart based on the average no of nonconformities per inspection unit. m
ci ui ; ni
ui :
u
i 1
average number of nonconformities per unit in a sample. ci: number of nonconformities in sample i (n is not necessary be integer ni: size of sample i
Example :
A personal computer manufacturer wishes to establish a control chart for nonconformities per unit on the final assembly line. The sample size is selected to be 5 computers. Data was collected in the following table for 20 samples.
Sample No. I
1 2 3 4
Sample size n
5 5 5 5
5
6 7 8 9 10
5
5 5 5 5 5
10
16 11 7 10 15
2.0
3.2 2.2 1.4 2.0 3.0
11
12 13 14 15 16
5
5 5 5 5 5
9
5 7 11 12 6
1.8
1.0 1.4 2.2 2.4 1.2
17
18 19 20
5
5 5 5
8
10 7 5
1.6
2.0 1.4 1.0
= 38.60/20 = 1.93
m
u n u n
LCL u 3 CL u UCL u 3
_ U=1.93
0 1 3 5 7 9 11 Sample 13 15 17 19
LCL=0.066
The preliminary data dont exhibit lack of statistical control , The trial control limits above would be adopted for current control purposes. the process is in control on this level. Although the process is in control, the average number of nonconformities per unit is still unacceptably high. Management action is necessary to improve the process.
Session 2.2 :
Control charts for Attributes with variable sample size
In some applications of the control chart for the fraction nonconforming, the sample is a 100% inspection of the process output over some period of time. Since different numbers of units could be produced in each period, the control chart would then have a variable sample size.
Variable Width Control Limits Control Limits Based on Average Sample Size Standardized Control Chart
Determine control limits for each individual sample that are based on the specific sample size. The upper and lower control limits are
p(1 p) p3 ni
Control charts based on the average sample size results in an approximate set of control limits. The average sample size is given by
n i 1 m
ni
p (1 p ) p3 n
The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties: Centerline at 0 UCL = 3 LCL = -3 The points plotted are given by:
Example
Sample No. i
1 2 3 4 5 6 7 8 9 10 11 12
Sample size ni
100 80 80 100 110 110 100 100 90 90 110 120
13
14 15 16 17 18 19 20 21 22 23 24 25
120
120 110 80 80 80 90 100 100 100 100 90 90
9
8 6 8 10 7 5 8 5 8 10 6 9
Solution
Sample No. i
1 2 3 4 5 6 7 8 9 10 11 12
ni
100 80 80 100 110 110 100 100 90 90 110 120
Di
12 8 6 9 10 12 11 16 10 6 20 15
Di / ni
0.120 0.100 0.075 0.090 0.091 0.109 0.110 0.160 0.110 0.067 0.182 0.125
zi
0.029 0.033 0.033 0.029 0.028 0.028 0.029 0.029 0.031 0.031 0.028 0.027
UCL
0.009 0 0 0.009 0.012 0.012 0.009 0.009 0.003 0.003 0.012 0.015
LCL
0.1183 0.195 0.195 0.183 0.180 0.180 0.183 0.183 0.189 0.189 0.180 0.177
13
14 15 16 17 18 19 20 21 22 23 24 25
120
120 110 80 80 80 90 100 100 100 100 90 90
9
8 6 8 10 7 5 8 5 8 10 6 9
0.075
0.067 0.055 0.100 0.125 0.088 0.056 0.080 0.050 0.080 0.100 0.067 0.100
0.027
0.027 0.028 0.033 0.033 0.033 0.031 0.029 0.029 0.029 0.029 0.031 0.031
0.015
0.015 0.012 0 0 0 0.003 0.009 0.009 0.009 0.009 0.003 0.003
0.177
0.177 0.180 0.195 0.195 0.195 0.189 0.183 0.183 0.183 0.183 0.189 0.189
UCL=0.1885
0.15
Proportion
0.10
_ P=0.0955
0.05
0.00 1 3 5 7 9 11 13 15 Sample 17 19 21 23 25
LCL=0.0026
0.20 UCL=0.1846
0.15
Proportion
0.10
_ P=0.0955
0.05
0.00 1 3 5 7 9 11 13 15 Sample 17 19 21 23 25
LCL=0.0064
Notes on control charts with variable sample size : We must be careful in analyzing runs or abnormal patterns on control charts with variable sample sizes. The problem is that a change in the sample fraction nonconforming must be interpreted relative to the sample size. Example: p = 0.2 p1 = 0.28 p2 = 0.24 n1 = 50 n2 = 250 1 = 1.41 2 = 1.58 It is clear that looking for runs or other random patterns is virtually meaningless here.
Tests
for runs and pattern could safely be applied to the standardized control charts.
The
difficulty in the standardized control chart is large for operating personal to understand and interpret. As the actual fraction nonconforming has been lost.
The
standardized control charts is also recommended when the length of production runs is short.
In
Control charts for nonconformities with variable sample size, it will be very difficult to use these procedures with c chart because both the center line and control limits will vary with the sample size.
The
Process Capability
Tolerances
design
Process
Once
a process is stable, the next emphasis is to ensure that the process is capable. Process capability refers to the ability of a process to produce a product that meets specifications.
The
again, a process is capable if individual products consistently meet specifications. A process is stable if only common variation is present in the process.
Design Specifications (a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time. Process Design Specifications (b) Design specifications and natural variation the same; process is capable of meeting specifications most of the time.
Process
Design Specifications (c) Design specifications greater than natural variation; process is capable of always conforming to specifications. Process Design Specifications (d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification.
Process
Out of Control
Capability
Not Capable