1. SIX SIGMA QUALITY METRICS vs TAGUCHI LOSS FUNCTION Luis Arimany de Pablos, Ph.D. www.calidad-seis-sigma.com

2. 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

3. 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 )

4. 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

5. 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 )

6. 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)

7. 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

14. Process Capability Index, Cp (Potential Capability) Cp = ( USL-LSL)/6  USL-LSL = Specification interval 6  = Process Capability

15. Process Centred at Target Process CpLSLUSL Right hand ppm defective 11 22 33 44 55 6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-2  2 m+2  2 m-3  m+3  m-4  4 m+4  4 m-5  5 m+5  5 m-6  6 m+6  6

16. 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

17. 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)?

18. 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

19. 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 

20. Process Capability Index, Cpk Cpk = ( USL-mp)/3  USL = Upper Specification Limit mp = process mean 3  =Half Process Capability

21. Process Centred at m + 1.5  ProcessCpk USL Right hand ppm defective 11 22 33 44 55 6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+2  2 0.5 m+3  1.5 m+4  4 2.5 m+5  5 3.5 m+6  6 4.5 Z score

22. Process Centred at m + 1.5  Process Right hand ppm defective 11 22 33 44 55 6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

23. QUALITY The Loss that a product or service produces to Society, in its production, transportation, consumption or use and disposal (Dr. Genichi Taguchi)

24. L=k(x i -m) 2 E(L)=k  2

25. Loss Function (Process Centred at Target) Six Sigma Metric Cp R H ppm defective 11 22 33 44 55 6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3 1.5   0.75  0.6  0.5  Standard Deviation 9k  2 2.25k  2 1k  2 0.56k  2 0.36k  2 0.25k  2

26. Loss Function (Process Centred at m+1.5  ) Six Sigma Metric Cpk R H ppm defective 11 22 33 44 55 6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3 1.5   0.75  0.6  0.5  Standard Deviation 29.25k  2 7.3125k  2 3.25k  2 1.8281k  2 1.17k  2 0.8125k  2

27. Six Sigma Metric Cpk 11 22 33 44 55 66 -0.16 0.16 0.5 0.83 1.16 1.5 Loss Function (Process Centred at m+1.5  ) 29.25k  2 7.3125k  2 3.25k  2 1.8281k  2 1.17k  2 0.8125k  2 Loss Function (Process Centred at m) 9k  2 2.25k  2 1k  2 0.56k  2 0.36k  2 0.25k  2 Cp 0.33 0.66 1 1.33 1.66 2

28. Six Sigma Metric Cpk 11 22 33 44 55 6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

29. AVERAGE RUN LENGTH 3 Sigma process Probability to detect the change 0.5 Average Run Length 2

30. AVERAGE RUN LENGTH 4 Sigma process Probability to detect the change 0.158655 Average Run Length 6.42

31. AVERAGE RUN LENGTH 5 Sigma process Probability to detect the change 0.02275 Average Run Length 43.45

32. AVERAGE RUN LENGTH 6 Sigma process Probability to detect the change 0.001349 Average Run Length 740.76

33. Six Sigma Metric Standard Deviation 33 44 55 66 33 0.75  3 0.6  3 0.5  3 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 USL

34. Six Sigma Metric Standard Deviation 33 44 55 66 33 0.75  3 0.6  3 0.5  3 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