Six Sigma Green Belt
Part 5
Process Capability
© 2011 IIE and Aft Systems, Inc.
5-1
Process Capability
•
•
Is the measured, inherent reproducibility of the
product turned out by the process. It can be
quantified from data which, in turn, are the result of
measurements of work performed by the process. It
defines limits we would normally expect virtually all
individuals to fall within.
By definition a process is “capable” when it is
operating at the three sigma level.
© 2011 IIE and Aft Systems, Inc.
5-2
Process Capability
•
•
Is the range over which the natural variation of a
process occurs as determined by the system of
common causes.
It is the ability of the combination of people,
machines, methods, materials, and measurements to
produce a product or service that will consistently
meet design specifications.
© 2011 IIE and Aft Systems, Inc.
5-3
Measuring Process Capability
•
•
Process capability is measured by the proportion
of output that can be produced within design
specifications-It is a long term prediction.
It is a measure of the uniformity of the process.
It can be measured only if all special causes
have been eliminated and the process is in a
state of statistical control.
© 2011 IIE and Aft Systems, Inc.
5-4
Components of Process Capability
•
•
•
Design specifications
Centering of natural variation
Range or spread of variation
© 2011 IIE and Aft Systems, Inc.
5-5
Capability Measures
•
•
Short term measures show the capability at a
specific instance in time, e.g., 5 out of 90 samples
did not meet customer requirements
Long term measures show the expected
capability of the process based on statistical
projections using inherent process variability
© 2011 IIE and Aft Systems, Inc.
5-6
Establishing Process Capability
•
•
Histogram
Control Chart
These must be performed prior to calculating any
process capability measures
© 2011 IIE and Aft Systems, Inc.
5-7
Statistics
Green Belts are expected to be able to calculate
the following:
•
•
•
Percent or proportion Non-Conforming
Cp Index
Cpk Index
© 2011 IIE and Aft Systems, Inc.
5-8
Percent Non-Conforming
•
•
Reflects the proportion of the population that we
normally expect not to meet the process
specifications.
Corresponds to the
‘tail’ areas on the
normal curve sketch
USL
LSL
LSL
Acceptable
© 2011 IIE and Aft Systems, Inc.
5-9
Calculating
Use z transform. Values in Normal Curve Table
zu 
zl 
Uspec - x

x is also called the mean
Lspec - x

R MR
 '  or
d2
d2
Process must be centered somewhere
between the customer requirements.
© 2011 IIE and Aft Systems, Inc.
5-10
Z Value
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.30
Tail
Prop.
0.4960
0.4920
0.4880
0.4840
0.4801
0.4761
0.4721
0.4681
0.4641
0.4602
0.4562
0.4522
0.4483
0.4443
0.4404
0.4364
0.4325
0.4286
0.4247
0.4207
0.4168
0.4129
0.4090
0.4052
0.4013
0.3974
0.3936
0.3897
0.3859
0.3821
Z Value
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.50
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
Tail Z Value
Prop.
0.3783
0.61
0.3745
0.62
0.3707
0.63
0.3669
0.64
0.3632
0.65
0.3594
0.66
0.3557
0.67
0.3520
0.68
0.3483
0.69
0.3446
0.70
0.3409
0.71
0.3372
0.72
0.3336
0.73
0.3300
0.74
0.3264
0.75
0.3228
0.76
0.3192
0.77
0.3156
0.78
0.3121
0.79
0.3085
0.80
0.3050
0.81
0.3015
0.82
0.2981
0.83
0.2946
0.84
0.2912
0.85
0.2877
0.86
0.2843
0.87
0.2810
0.88
0.2776
0.89
0.2743
0.90
Tail Z Value
Prop.
0.2709
0.91
0.2676
0.92
0.2643
0.93
0.2611
0.94
0.2578
0.95
0.2546
0.96
0.2514
0.97
0.2483
0.98
0.2451
0.99
0.2420
1.00
0.2389
1.01
0.2358
1.02
0.2327
1.03
0.2296
1.04
0.2266
1.05
0.2236
1.06
0.2206
1.07
0.2177
1.08
0.2148
1.09
0.2119
1.10
0.2090
1.11
0.2061
1.12
0.2033
1.13
0.2005
1.14
0.1977
1.15
0.1949
1.16
0.1922
1.17
0.1894
1.18
0.1867
1.19
0.1841
1.20
© 2011 IIE and Aft Systems, Inc.
Tail
Prop.
0.1814
0.1788
0.1762
0.1736
0.1711
0.1685
0.1660
0.1635
0.1611
0.1587
0.1562
0.1539
0.1515
0.1492
0.1469
0.1446
0.1423
0.1401
0.1379
0.1357
0.1335
0.1314
0.1292
0.1271
0.1251
0.1230
0.1210
0.1190
0.1170
0.1151
5-11
Z Value
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
Tail Z Value
Prop.
0.1131 1.51
0.1112 1.52
0.1093 1.53
0.1075 1.54
0.1056 1.55
0.1038 1.56
0.1020 1.57
0.1003 1.58
0.0985 1.59
0.0968 1.60
0.0951 1.61
0.0934 1.62
0.0918 1.63
0.0901 1.64
0.0885 1.65
0.0869 1.66
0.0853 1.67
0.0838 1.68
0.0823 1.69
0.0808 1.70
0.0793 1.71
0.0778 1.72
0.0764 1.73
0.0749 1.74
0.0735 1.75
0.0721 1.76
0.0708 1.77
0.0694 1.78
0.0681 1.79
0.0668 1.80
Tail Z Value
Prop.
0.0655 1.81
0.0643 1.82
0.0630 1.83
0.0618 1.84
0.0606 1.85
0.0594 1.86
0.0582 1.87
0.0571 1.88
0.0559 1.89
0.0548 1.90
0.0537 1.91
0.0526 1.92
0.0516 1.93
0.0505 1.94
0.0495 1.95
0.0485 1.96
0.0475 1.97
0.0465 1.98
0.0455 1.99
0.0446 2.00
0.0436 2.01
0.0427 2.02
0.0418 2.03
0.0409 2.04
0.0401 2.05
0.0392 2.06
0.0384 2.07
0.0375 2.08
0.0367 2.09
0.0359 2.10
Tail Z Value
Prop.
0.0351 2.11
0.0344 2.12
0.0336 2.13
0.0329 2.14
0.0322 2.15
0.0314 2.16
0.0307 2.17
0.0301 2.18
0.0294 2.19
0.0287 2.20
0.0281 2.21
0.0274 2.22
0.0268 2.23
0.0262 2.24
0.0256 2.25
0.0250 2.26
0.0244 2.27
0.0239 2.28
0.0233 2.29
0.0228 2.30
0.0222 2.31
0.0217 2.32
0.0212 2.33
0.0207 2.34
0.0202 2.35
0.0197 2.36
0.0192 2.37
0.0188 2.38
0.0183 2.39
0.0179 2.40
© 2011 IIE and Aft Systems, Inc.
Tail
Prop.
0.0174
0.0170
0.0166
0.0162
0.0158
0.0154
0.0150
0.0146
0.0143
0.0139
0.0136
0.0132
0.0129
0.0125
0.0122
0.0119
0.0116
0.0113
0.0110
0.0107
0.0104
0.0102
0.0099
0.0096
0.0094
0.0091
0.0089
0.0087
0.0084
0.0082
5-12
Z Value
2.41
2.42
2.43
2.44
2.45
2.46
2.47
2.48
2.49
2.50
2.51
2.52
2.53
2.54
2.55
2.56
2.57
2.58
2.59
2.60
2.61
2.62
2.63
2.64
2.65
2.66
2.67
2.68
2.69
2.70
Tail Z Value
Prop.
0.0080 2.71
0.0078 2.72
0.0075 2.73
0.0073 2.74
0.0071 2.75
0.0069 2.76
0.0068 2.77
0.0066 2.78
0.0064 2.79
0.0062 2.80
0.0060 2.81
0.0059 2.82
0.0057 2.83
0.0055 2.84
0.0054 2.85
0.0052 2.86
0.0051 2.87
0.0049 2.88
0.0048 2.89
0.0047 2.90
0.0045 2.91
0.0044 2.92
0.0043 2.93
0.0041 2.94
0.0040 2.95
0.0039 2.96
0.0038 2.97
0.0037 2.98
0.0036 2.99
0.0035 3.00
Tail Z Value
Prop.
0.0034 3.10
0.0033 3.20
0.0032 3.30
0.0031 3.40
0.0030 3.50
0.0029 3.60
0.0028 3.70
0.0027 3.80
0.0026 3.90
0.0026 4.00
0.0025 4.10
0.0024 4.20
0.0023 4.30
0.0023 4.40
0.0022 4.50
0.0021 4.60
0.0021 4.70
0.0020 4.80
0.0019 4.90
0.0019 5.00
0.0018 5.10
0.0018 5.20
0.0017 5.30
0.0016 5.40
0.0016 5.50
0.0015 6.00
0.0015 6.50
0.0014 7.00
0.0014 7.50
0.0013 8.00
Tail
Prop.
0.00096760
0.00068714
0.00048342
0.00033693
0.00023263
0.00015911
0.00010780
0.00007235
0.00004810
0.00003167
0.00002066
0.00001335
0.00000854
0.00000541
0.00000340
0.00000211
0.00000130
0.00000079
0.00000048
0.00000029
0.00000017
0.00000010
0.00000006
0.00000003
0.00000002
0.000000001
0.00000000
0.00000000
0.00000000
0.00000000
© 2011 IIE and Aft Systems, Inc.
5-13
Example
1.
2.
3.
A process has a mean of
600 and a standard
deviation of 6. Spec
limits are 585 and 615.
What is the percent nonconforming? (The total
proportion in both tails.)
What is the DPMO?
What is the sigma level?
(The sigma level is the
smaller of the two z
values.)
Unacceptable
LSL
LSL
© 2011 IIE and Aft Systems, Inc.
USL
Unacceptable
5-14
Example
Unacceptable
LSL
LSL
USL
1.
Unacceptable
2.
3.
A process has a mean of
100 and a standard
deviation of 4. Spec
limits are 95 and 106.
What is the percent nonconforming?
What is the sigma level?
What is the DPM?
© 2011 IIE and Aft Systems, Inc.
5-15
Example
Continued
In the preceding example if each incorrect
transaction costs $4 and annual volume is
400,000, what is the cost of the incorrect
transactions?
© 2011 IIE and Aft Systems, Inc.
5-16
Example
Continued
If in the preceding example the standard
deviation were reduced to 2 via a DMAIIC
Green Belt project. If this were done at a
cost of $100,000 would it be worth it? (One year
payback.)
© 2011 IIE and Aft Systems, Inc.
5-17
Example
Continued
What would it take (or be worth) to now reduce
the standard deviation to 1?
© 2011 IIE and Aft Systems, Inc.
5-18
More Practice
•
•
•
A process has a mean of
200 and and a standard
deviation of 2.
Upper specification limit
is 212.
Lower specification limit
is 188.
•
Determine
–
–
–
Proportion non
conforming
DPMO
Sigma Level
Motorola Shift
© 2011 IIE and Aft Systems, Inc.
5-19
Another Problem
What is the capability of the process, in terms of
proportion defective or sigma level or dpmo, we
collected data on earlier in the course?
(page 4-12)
© 2011 IIE and Aft Systems, Inc.
5-20
Capability Indices
Show the relationship between the process capability and
the process specifications
Cp measures potential capability assuming that
the process average is equal to the midpoint of
the specification limits and the process is
operating in statistical control. Cpk reflects the
current process mean’s proximity to either
specification limit. (When the process is centered
Cp = Cpk.)
Although the indices are calculated differently,
the interpretation is the same
© 2011 IIE and Aft Systems, Inc.
5-21
Cp or Cpk Less than 1.0
Cp or Cpk
Value
Sigma
Level
0.33
0.67
1
2
© 2011 IIE and Aft Systems, Inc.
5-22
4.39
4.15
3.91
3.67
3.43
3.19
2.95
2.71
2.47
2.23
1.99
1.75
1.51
1.27
1.03
0.79
0.55
0.31
0.07
-0.2
-0.4
-0.7
-0.9
-1.1
-1.4
-1.6
-1.9
-2.1
-2.3
-2.6
-2.8
-3.1
-3.3
-3.5
-3.8
-4
-4.3
-4.5
Cp or Cpk Equal to 1.0
Cp or Cpk
Value
Sigma
Level
0.33
0.67
1.00
1
2
3
© 2011 IIE and Aft Systems, Inc.
5-23
4.39
4.15
3.91
3.67
3.43
3.19
2.95
2.71
2.47
2.23
1.99
1.75
1.51
1.27
1.03
0.79
0.55
0.31
0.07
-0.2
-0.4
-0.7
-0.9
-1.1
-1.4
-1.6
-1.9
-2.1
-2.3
-2.6
-2.8
-3.1
-3.3
-3.5
-3.8
-4
-4.3
-4.5
Cp or Cpk Greater than 1.0
Cp or Cpk
Value
Sigma
Level
0.33
0.67
1.00
1.33
1.67
2.00
2.33
2.67
3.00
1
2
3
4
5
6
7
8
9
No Motorola Shift
© 2011 IIE and Aft Systems, Inc.
5-24
Calculating Cp
Cp = (Uspec – Lspec)/6
Process must be centered at the midpoint of the
specification limits.
© 2011 IIE and Aft Systems, Inc.
5-25
Cpk Calculation
Smaller of the following:
Cpku = (Uspec – Mean) / 3
Cpkl = (Mean – Lspec) / 3
Cpk also is equal to the sigma level divided
by 3
Note:
Can also use these for one sided requirements
© 2011 IIE and Aft Systems, Inc.
5-26
Capability
What is the capability index for the process we
analyzed earlier? (page 4-12)
© 2011 IIE and Aft Systems, Inc.
5-27
Interpretation
Capability
Measure
Not
Capable
Proportion
Non
More than
Conforming
.0027
More than
DPMO
2700
Less than
Sigma Level
3.0
Less than
Cpk
1.0
Capable
More
than
Capable
0.0027
Less than
.0027
2700
3.0
1
© 2011 IIE and Aft Systems, Inc.
Less than
2700
More
than 3.0
More
than 1.0
5-28
Another Cost Study
•
•
•
•
A process has customer requirements of 50 and 75. It is in
control with an adjustable mean and a known standard
deviation (based on the Shewhart approximation) of 8.
Accounting tells us that a “defect” on the low side will
cost of $10 and on the high side $5. (Scrap vs. rework.)
Engineering has not been able to figure out a way to
reduce variation.
What is the optimum location for the process centering in
order to minimize cost?
© 2011 IIE and Aft Systems, Inc.
5-29
Homework Exercise
U-Bolts
•
•
On the following page is
data on U-bolts for the
dimension shown at the
right.
Determine the capability of
the process regarding the
indicated dimension. (Specs
are 10.70 + .15)
© 2011 IIE and Aft Systems, Inc.
5-30
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
10.65
10.75
10.75
10.60
10.70
10.60
10.75
10.65
10.60
10.80
10.85
10.70
10.65
10.70
10.65
10.75
10.90
10.75
10.75
10.65
10.60
10.50
10.80
10.65
10.70
10.85
10.80
10.70
10.75
10.75
10.80
10.80
10.70
10.75
10.70
10.75
10.70
10.80
10.80
10.70
10.70
10.65
10.60
10.55
10.65
10.60
10.70
10.65
Observations
10.65
10.75
10.80
10.65
10.70
10.75
10.65
10.50
10.70
10.60
10.65
10.75
10.75
10.90
10.65
10.65
10.80
10.70
10.75
10.70
10.65
10.70
10.65
10.90
10.80
10.70
10.70
10.65
10.65
10.75
10.75
10.70
10.65
10.75
10.70
10.70
10.65
10.80
10.60
10.75
10.70
10.65
10.60
10.70
10.70
10.65
10.50
10.80
© 2011 IIE and Aft Systems, Inc.
10.55
10.65
10.75
10.90
10.65
10.70
10.45
10.80
10.60
10.65
10.65
10.65
10.50
10.70
10.65
10.65
10.75
10.55
10.90
10.65
10.65
10.75
10.65
10.65
Average
10.670
10.740
10.760
10.700
10.690
10.710
10.680
10.690
10.650
10.710
10.710
10.710
10.650
10.780
10.670
10.700
10.770
10.660
10.720
10.650
10.650
10.640
10.660
10.730
Range
0.25
0.20
0.10
0.30
0.10
0.15
0.35
0.30
0.10
0.20
0.20
0.10
0.25
0.20
0.20
0.10
0.20
0.20
0.30
0.15
0.10
0.25
0.30
0.25
104
5-31
Statistics
Mean
10.6958
Average Range
.2021
Shewhart Standard Deviation .0869
Calculated Standard
Deviation
.0871
© 2011 IIE and Aft Systems, Inc.
5-32
28
27
26
25
24
23
22
21
20
19
18
17
16
Frequency 15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
© 2011 IIE and Aft Systems, Inc.
5-33
Sample ID
Averages
Ranges
© 2011 IIE and Aft Systems, Inc.
5-34
Histogram
40
35
# Observations
30
25
20
15
10
5
0
10.45 10.495 10.54 10.585 10.63 10.675 10.72 10.765 10.81 10.855
to <=
to <=
to <=
to <=
to <=
to <=
to <=
to <=
to <=
to <=
10.495 10.54 10.585 10.63 10.675 10.72 10.765 10.81 10.855 10.9
Class
© 2011 IIE and Aft Systems, Inc.
5-35
Xbar Chart
10.85
10.8
10.75
10.7
10.65
10.6
10.55
10.5
10.45
UCL=10.81244
CEN=10.69583
LCL=10.57923
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
19
20
21
22
23
24
R Chart
0.5
UCL=0.4272
0.4
0.3
CEN=0.20208
0.2
0.1
0
1
LCL=0.0
2
3
4
5
6
7
8
9
10
11
12
13
© 2011 IIE and Aft Systems, Inc.
14
15
16
17
18
5-36
© 2011 IIE and Aft Systems, Inc.
11.07
11.05
11.02
11.00
10.97
10.95
10.93
10.90
10.88
10.85
10.83
10.80
10.78
10.75
10.73
10.70
10.68
Mean = 10.696
StdDev = 0.0869
USL = 10.85
LSL = 10.55
Sigma Level = 1.6801
10.65
10.63
10.60
10.58
10.55
10.53
10.50
10.48
10.45
10.43
10.40
10.38
10.35
10.33
10.30
Process Capability
Cpk Analysis
Cpk = .5600
DPM = 84,654
5-37