six sigma quality metrics vs taguchi loss function luis arimany de pablos, ph.d
TRANSCRIPT
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SIX SIGMA QUALITY METRICS
vs
TAGUCHI LOSS FUNCTION
Luis Arimany de Pablos, Ph.D.
www.calidad-seis-sigma.com
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outside the mean 2 a maximum 25% of the values
outside the mean 3 a maximum 11.11% of the values
outside the mean 4 a maximum 6.25% of the values
outside the mean 5 a maximum 4% of the values
outside the mean 6 a maximum 2.77% of the values
FOR ANY DISTRIBUTION
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outside the mean 2 there are 4.55% of the values
outside the mean 3 there are 0.27% of the values
outside the mean 4 there are 0.006% of the values
outside the mean 5 there are 5.74·10-5% of the values
outside the mean 6 there are 19.8·10-8 % of the values
FOR NORMAL DISTRIBUTION
( two tails )
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One of Motorola´s most significant contributions was to change the discussion of quality, from quality levels measured in % (parts-per-
hundred), to one, in parts per million, or, even, parts per
billion
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to the right of the mean + 2 there are 22,750 per million
to the right of the mean +3 there are 1,349.96 per million
to the right the mean + 4 there are 31.686 per million
to the right of the mean + 5 there are 0.28715 per million
to the right of the mean + 6 there are 0.001 per million
FOR NORMAL DISTRIBUTION
( one tail )
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DEFECTIVE PRODUCT OR SERVICE
X USLX LSL
If we set the Specification Limits at m 3
On average 0.27 % defectives
2.7 per thousand
2,700 per million
1,350 per million (one tail)
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We should have a process with such a low dispersion that Specification Limits are at:
m 6
0.00198 defective per million
0.001 per million in one tail
0.002 per million
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1 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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2 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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3 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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4 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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5 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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6 sigma process
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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Process Capability Index, Cp(Potential Capability)
Cp = ( USL-LSL)/6
USL-LSL = Specification interval
6 = Process Capability
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Process Centred at Target
Process Cp LSL USLRight hand ppm
defective
1
23
4
5
6
158,655
22,750
1,350
31.686
0.287
0.001
0.33
0.66
1
1.33
1.66
2
m-1 m+1
m-22 m+22
m-3 m+3
m-44 m+44
m-55 m+55
m-66 m+66
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We should have a process with such a low dispersion that Specification Limits are at:
m 6
0.00198 defective per million
0.001 per million in one tail
0.002 per million
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Working with 6 methodology you get
3.4 defectives per million
How can this be, if the exact figure is 0.002 ppm
(or 0.001 ppm if we consider only one tail)?
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Even if a process is under control it is not infrequent to see that the process mean moves up (or down) to target
mean plus (minus) 1.5 .
If this is the case, the worst case, working with the 6 Philosophy will guarantee that we will not get more
than 3.4 defectives per million products or services
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Let us assume that the process mean is not at the mid-point of the
specification interval, the target value m, but at
m+1.5
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Process Capability Index, Cpk
Cpk = ( USL-mp)/3
USL = Upper Specification Limit
mp = process mean
3 =Half Process Capability
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Process Centred at m + 1.5
Process Cpk USL Right hand ppm
defective
1
23
4
5
6
691,464
308,536
66,807
6,209.66
232.67
3.4
-0.166
0.166
0.5
0.83
1.166
1.5
m+1-0.51
m+220.5
m+31.5
m+442.5
m+553.5
m+664.5
Z score
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Process Centred at m + 1.5
ProcessRight hand ppm
defective
1
23
4
5
6
691,464
308,536
66,807
6,209.66
232.67
3.4
Process Centred at m
Cpk
-0.166
0.166
0.5
0.83
1.166
1.5
Right hand ppm defective
Cp
0.33
0.66
1
1.33
1.66
2
158,655
22,750
1,350
31.69
0.287
0.001
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QUALITY
The Loss that a product or service produces to Society, in its production, transportation,
consumption or use and disposal
(Dr. Genichi Taguchi)
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L=k(xi-m)2
E(L)=k2
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Loss Function(Process Centred at Target)
Six Sigma
MetricCp
R H ppm defective
1
23
4
5
6
158,655
22,750
1,350
31.686
0.287
0.001
0.33
0.66
1
1.33
1.66
2
Loss Function
3
1.5
0.75
0.6
0.5
Standard Deviation
9k2
2.25k2
1k2
0.56k2
0.36k2
0.25k2
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Loss Function(Process Centred at m+1.5)
Six Sigma
MetricCpk
R H ppm defective
1
23
4
5
6
691,464
308,536
66,807
6,209.66
232.67
3.4
-0.16
0.16
0.5
0.83
1.16
1.5
Loss Function
3
1.5
0.75
0.6
0.5
Standard Deviation
29.25k2
7.3125k2
3.25k2
1.8281k2
1.17k2
0.8125k2
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Six Sigma
MetricCpk
1
23
4
5
6
-0.16
0.16
0.5
0.83
1.16
1.5
Loss Function
(Process Centred at m+1.5)
29.25k2
7.3125k2
3.25k2
1.8281k2
1.17k2
0.8125k2
Loss Function
(Process Centred at m)
9k2
2.25k2
1k2
0.56k2
0.36k2
0.25k2
Cp
0.33
0.66
1
1.33
1.66
2
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Six Sigma
MetricCpk
1
23
4
5
6
-0.16
0.16
0.5
0.83
1.16
1.5
R H ppm defective
(Process Centred at m+1.5)
R H ppm
defective (Process Centred at
m)Cp
0.33
0.66
1
1.33
1.66
2
158,655
22,750
1,350
31.686
0.287
0.001
691,464
308,536
66,807
6,209.66
232.67
3.4
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AVERAGE RUN LENGTH
3 Sigma process
Probability to detect the change
0.5
Average Run Length
2
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AVERAGE RUN LENGTH
4 Sigma process
Probability to detect the change
0.158655
Average Run Length
6.42
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AVERAGE RUN LENGTH
5 Sigma process
Probability to detect the change
0.02275
Average Run Length
43.45
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AVERAGE RUN LENGTH
6 Sigma process
Probability to detect the change
0.001349
Average Run Length
740.76
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Six Sigma
MetricStandard Deviation
3
4
5
6
3
0.753
0.63
0.53
Probability of Defectives
after the Shift
Expected Number of samples to
detect the Shift
2
6.42
43.45
740.76
0.5
0.158655
0.02275
0.001349
Average Run Length
n/3σmn/3σm 33
n/3σmn/σ4m 34
n/3σmn/σ5m 35
n/3σmn/σ6m 36
USL
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Six Sigma
Metric
Standard Deviation
3
4
5
6
3
0.753
0.63
0.53
Probability of Defectives after the
Shift
Expected Number of samples to
detect the Shift
2
6.42
43.45
740.76
0.5
0.158655
0.02275
0.00134996
Average Run Length