SIX SIGMA QUALITY METRICS vs TAGUCHI LOSS FUNCTION Luis Arimany de Pablos, Ph.D. .

SIX SIGMA QUALITY METRICS

vs

TAGUCHI LOSS FUNCTION

Luis Arimany de Pablos, Ph.D.

www.calidad-seis-sigma.com

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

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 )

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

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 )

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)

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

1 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

2 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

3 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

4 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

5 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

6 sigma process

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Process Capability Index, Cp(Potential Capability)

Cp = ( USL-LSL)/6

USL-LSL = Specification interval

6 = Process Capability

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

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

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)?

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

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

Process Capability Index, Cpk

Cpk = ( USL-mp)/3

USL = Upper Specification Limit

mp = process mean

3 =Half Process Capability

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

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

QUALITY

The Loss that a product or service produces to Society, in its production, transportation,

consumption or use and disposal

(Dr. Genichi Taguchi)

L=k(xi-m)2

E(L)=k2

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

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

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

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

AVERAGE RUN LENGTH

3 Sigma process

Probability to detect the change

0.5

Average Run Length

2

AVERAGE RUN LENGTH

4 Sigma process

Probability to detect the change

0.158655

Average Run Length

6.42

AVERAGE RUN LENGTH

5 Sigma process

Probability to detect the change

0.02275

Average Run Length

43.45

AVERAGE RUN LENGTH

6 Sigma process

Probability to detect the change

0.001349

Average Run Length

740.76

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

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