Unformatted text preview: Chapter 3
Chapter
Statistical Process
Control Statistical Process
Control
Control • Take periodic samples from
Take
process
process
• Plot sample points on a
Plot
control chart
control
• Determine if process UCL
Determine
is within limits
is
• Correct process before
Correct
LCL
defects are produced
defects Control Chart
Control
Out of control Upper
control
limit Process
average Lower
control
limit 1 2 3 4 5 6 Sample number 7 8 9 10 Variation
Variation
Common Causes
Common
Variation inherent in a process
Can be eliminated only through
improvements in the system Special Causes
Variation due to identifiable factors
Can be modified through operator or
management action Types of Data
Types
Attribute data
Product characteristic
evaluated with a discrete choice
Good/bad, yes/no Variable data
Product characteristic that
can be measured
Length, size, weight, height,
time, velocity Control Charts for
Attributes
Attributes
p Charts
Calculate percent defectives in sample
Based on binonomial distribution c Charts
Count number of defects in an item Based on Poisson distribution
Based pChart
pChart
UCL = p + zσ p
UCL
where LCL = p  zσ p
LCL
z
=
the number of standard
the
deviations from the process average
deviations
p
=
the sample percent
the
defective; an estimate of the process
average
average
σp
=
the standard deviation of
the
the sample proportion
the
p(1  p)
(1
σp =
n pChart Example
pChart
SAMPLE 1
2
3
:
:
20 NUMBER OF
DEFECTIVES 6
0
4
:
:
18
Total 200 PROPORTION
DEFECTIVE 6/100 = .06
6/
.00
.04
:
:
.18
Avg .10 20 samples of 100 pairs of jeans
20
100 pChart Calculations
pChart
p= total defectives
= 200 / 20(100) = 0.10
total sample observations UCL = p + z p(1  p)
= 0.10 + 3
n 0.10(1  0.10)
100 UCL = 0.190
LCL = p  z
LCL = 0.010 p(1  p)
= 0.10  3
n 0.10(1  0.10)
100 0.20 Is the
process
in control? UCL = 0.190 0.18 Proportion defective 0.16
0.14
0.12
0.10 No! Pts
are outside
the limits. p = 0.10 0.08
0.06
0.04
0.02
0.02 LCL = 0.010
2 4 6 8
10
12 14
Sample number 16 18 20 cChart
cChart
UCL = c + zσ c
UCL
LCL = c  zσ c
LCL σc = c where
c = number of defects per sample cChart Example
cChart
The number of defects in 15 sample rooms
SAMPLE NUMBER OF DEFECTS 1
2
3 12
8
16 :
: :
: 15 15
190 cChart
cChart
24 UCL = 23.35 Yes Number of defects Is this
process
in control? 21
18 c = 12.67 15
12
9
6
LCL = 1.99 3 2 4 6 8 10 Sample number 12 14 16 Control Charts
for Variables
for Mean chart ( Xbar Chart ) uses average of a sample Range chart ( R Chart ) uses amount of dispersion in a
sample Xbar Chart
Xbar
= = x1 + x2 + ... xk
x
k
=+ A R
= A R
UCL = x
LCL = x
UCL
LCL
2
2
where
=
x = average of sample means
R = range
range
A2 = value from 3σ table Xbar Chart Example
Xbar
OBSERVATIONS (SLIP RING DIAMETER, CM)
SAMPLE k 1 2 3 4 5 x bar R 1
2
3
4
5
6
7
8
9
10 5.02
5.01
4.99
5.03
4.95
4.97
5.05
5.09
5.14
5.01 5.01
5.03
5.00
4.91
4.92
5.06
5.01
5.10
5.10
4.98 4.94
5.07
4.93
5.01
5.03
5.06
5.10
5.00
4.99
5.08 4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07 4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99 4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03 0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10 50.09 1.15 Total
Xbar is the average of each row. R is the range, i.e., the difference between the largest and smallest value in a row. X bar Chart
XCalculations
Calculations
50.09
= ∑x
x=
=
= 5.01 cm
10
k
Xbarbar is the average of the x’s
=
UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08
=
LCL = x  A2R = 5.01  (0.58)(0.115) = 4.94
Lookup A2 from the table (on next slide) by sample size
Look back at the data (on previous slide).
There are 10 samples. The size of each sample is 5. Determining Control Limits for xbar and RCharts
SAMPLE SIZE
n FACTOR FOR xCHART
A2 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.44
0.11
0.99
0.77
0.55
0.44
0.22
0.11
0.00
0.99
0.99
0.88 Fa
Fa
ct
or
s FACTORS FOR RCHART
D3
D4
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
0.36
0.38
0.39
0.40
0.41 3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
1.64
1.62
1.61
1.61
1.59 5.10 – Is this
process
in control? 5.08 – UCL = 5.08 5.06 – Yes, pt 9
is on the
line,
not over it. Mean 5.04 –
=
x = 5.01 5.02 –
5.00 –
4.98 –
4.96 – X bar
Chart
Chart LCL = 4.94 4.94 –
4.92 –

1 
2 
3 



4
5
6
7
Sample number 
8 
9 
10 R Chart
RUCL = D4R
UCL LCL = D3R
LCL ∑R
R=
k
where
R = range of each sample
k = number of samples
D3, D4 = values from table RChart Example
RChart
OBSERVATIONS (SLIPRING DIAMETER, CM)
SAMPLE k
SAMPLE 1 2 3 4 5 x R 1
2
3
4
5
6
7
8
9
10 5.02
5.01
4.99
5.03
4.95
4.97
5.05
5.09
5.14
5.01 5.01
5.03
5.00
4.91
4.92
5.06
5.01
5.10
5.10
4.98 4.94
5.07
4.93
5.01
5.03
5.06
5.10
5.00
4.99
5.08 4.99
4.95
4.92
4.98
5.05
4.96
4.96
4.99
5.08
5.07 4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99 4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03 0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10 50.09 1.15 Total
Example 15.3 RChart Calculations
RChart
∑R
1.15
R=
=
= 0.115
k
10 UCL = D4R = 2.11(0.115) = 0.243
LCL = D3R = 0(0.115) = 0 Lookup Values D3 and D4 in table
in Determining Control Limits for xbar and RCharts
SAMPLE SIZE
n FACTOR FOR xCHART
A2 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.44
0.11
0.99
0.77
0.55
0.44
0.22
0.11
0.00
0.99
0.99
0.88 Fa
Fa
ct
or
s FACTORS FOR RCHART
D3
D4
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
0.36
0.38
0.39
0.40
0.41 3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
1.64
1.62
1.61
1.61
1.59 RChart
RChart
0.28 –
0.24 – UCL = 0.243 Range 0.20 –
0.16 – R = 0.115 0.12 –
0.08 –
0.04 –
0– LCL = 0



1
2
3 



4
5
6
7
Sample number 
8 
9 
10 Use X bar and
RCharts Together
RCharts Process average and process variability must be
in control. It is possible for samples to have very narrow
ranges, but their averages is beyond control
limits. It is possible for sample averages to be in
control, but ranges might be very large. A Process is In Control if
Process
1. No sample points are outside the
No
control limits
control
2. Most points are near the process
Most
average
average
3. About an equal number of points
About
are above & below the centerline
are
4. Points appear to be randomly
Points
distributed
distributed Control Chart Patterns
Control
1. 8 consecutive points on one side of the
consecutive
center line.
center
2. 8 consecutive points up or down across
consecutive
zones.
zones.
3. 14 points alternating up and down.
4. 2 out of 3 consecutive points in Zone A
out
but still inside the control limits.
but
5. 4 out of 5 consecutive points in Zone A
out
or B.
or
You don’t have to memorize these. They will be provided on quiz. Zones for Pattern Tests
Zones Figure 15.4 Rule 1: Eight consecutive points
on one side of the center line UCL UCL Mean Mean LCL LCL Rule 2: Eight consecutive points
up or down across zones
UCL UCL LCL LCL Rule 3: Fourteen points
alternating up or down
alternating
UCL LCL Rule 4: Two of three
consecutive points in Zone A
consecutive
UCL UCL
Zone A
Zone B
Zone C
Zone C LCL Zone A
Zone B
Zone C
Zone C Zone B
Zone A Zone B
Zone A LCL Rule 5: 4 out of 5 consecutive
points in Zone B or beyond
points
UCL UCL Zone C Zone A
Zone B
Zone C Zone C
Zone B Zone C
Zone B Zone A Zone A Zone A
Zone B LCL LCL Process Capability
Process The range of natural variability in a
The
process
process A process cannot meet specifications
process
if its natural variability exceeds the
design tolerances
design
3sigma quality
Specifications equal the process control
limits. 6sigma quality
Specifications are twice as large as control
Specifications
limits
limits Process Capability
Process
Design
Design
Specifications
Specifications Cp < 1;
Process is
not capable (a) Natural variation
(a)
exceeds design
specifications; process
is not capable of
meeting specifications
all the time.
all
Process Design
Design
Specifications
Specifications
(b) Design specifications
b)
and natural variation the
same; process is capable
of meeting specifications
most the time.
most Cp = 1;
Process is just capable Process Process Capability
Process Cp > 1;
process is
capable Design
Design
Specifications
Specifications
(c) Design specifications
c)
greater than natural
variation; process is
capable of always
conforming to
specifications.
specifications.
Process
Design
Design
Specifications
Specifications
(d) Specifications greater
d)
than natural variation, but
process off center;
capable but some output
will not meet upper
specification.
specification. Cpk measures if process is offcenter Process If Cp = 2,
Six sigma
quality Process Capability
Process
Process Capability Ratio
tolerance range
Cp =
process range = upper specification limit upper
lower specification limit
lower
6σ where 6σ = UCL – LCL Process Capability Index, Cpk
Process Cpk = minimum =
x  lower specification limit
,
3σ
=
upper specification limit  x
3σ Recall, for xbar chart, A2(Rbar) = 3σ ...
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This note was uploaded on 08/30/2011 for the course ART 3514 taught by Professor Dhbannan during the Summer '03 term at Virginia Tech.
 Summer '03
 DHBannan

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