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SIX SIGMA QUALITY METRICS vs TAGUCHI LOSS FUNCTION

SIX SIGMA QUALITY METRICS vs TAGUCHI LOSS FUNCTION. Luis Arimany de Pablos, Ph.D. www.calidad-seis-sigma.com. FOR ANY DISTRIBUTION. outside the mean  2 a maximum 25% of the values. outside the mean  3 a maximum 11.11% of the values.

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SIX SIGMA QUALITY METRICS vs TAGUCHI LOSS FUNCTION

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

  2. FOR ANY DISTRIBUTION 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

  3. FOR NORMAL 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 ( 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. FOR NORMAL DISTRIBUTION 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 ( one tail )

  6. DEFECTIVE PRODUCT OR SERVICE X  LSL X  USL 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 0.002 per million 0.001 per million in one tail

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

  9. Process Centred at Target Right hand ppm defective Process Cp LSL USL 158,655 1 0.33 m-1 m+1 m-22 m+22 2 0.66 22,750 3 1 m-3 m+3 1,350 4 1.33 m-44 m+44 31.686 5 1.66 m-55 m+55 0.287 6 2 m-66 m+66 0.001

  10. We should have a process with such a low dispersion that Specification Limits are at: m 6  0.00198 defective per million 0.002 per million 0.001 per million in one tail

  11. 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)?

  12. 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

  13. 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

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

  15. Process Centred at m + 1.5  Right hand ppm defective Process Cpk USL Z score 691,464 1 m+1 -0.51 -0.166 m+22 0.5 2 0.166 308,536 3 m+3 1.5 66,807 0.5 4 m+44 2.5 6,209.66 0.83 5 m+55 3.5 232.67 1.166 6 m+66 4.5 3.4 1.5

  16. Process Centred at m + 1.5  Process Centred at m Right hand ppm defective Right hand ppm defective Cp Cpk Process 1 0.33 158,655 -0.166 691,464 2 308,536 0.66 22,750 0.166 3 1,350 0.5 66,807 1 4 6,209.66 1.33 31.69 0.83 5 232.67 1.66 0.287 1.166 6 2 0.001 1.5 3.4

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

  18. L=k(xi-m)2 E(L)=k2

  19. Loss Function (Process Centred at Target) R H ppm defective Six Sigma Metric Standard Deviation Loss Function Cp 1 3 0.33 158,655 9k2 2.25k2 2 1.5 0.66 22,750 3  1 1,350 1k2 4 0.75 1.33 31.686 0.56k2 5 0.6 0.36k2 1.66 0.287 6 0.5 0.001 0.25k2 2

  20. Loss Function (Process Centred at m+1.5) R H ppm defective Six Sigma Metric Standard Deviation Loss Function Cpk 1 3 -0.16 691,464 29.25k2 7.3125k2 2 1.5 0.16 308,536 3  0.5 66,807 3.25k2 4 0.75 0.83 6,209.66 1.8281k2 5 0.6 1.17k2 1.16 232.67 6 0.5 3.4 0.8125k2 1.5

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

  22. R H ppm defective (Process Centred at m) R H ppm defective (Process Centred at m+1.5) Six Sigma Metric Cp Cpk 1 0.33 -0.16 158,655 691,464 2 0.66 0.16 22,750 308,536 3 1 0.5 1,350 66,807 4 1.33 0.83 31.686 6,209.66 5 0.287 232.67 1.66 1.16 6 0.001 3.4 2 1.5

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

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

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

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

  27. Average Run Length Six Sigma Metric Expected Number of samples to detect the Shift Probability of Defectives after the Shift Standard Deviation USL 3 3 0.5 2 4 0.753 0.158655 6.42 5 0.63 0.02275 43.45 6 0.53 0.001349 740.76

  28. Average Run Length Expected Number of samples to detect the Shift Six Sigma Metric Standard Deviation Probability of Defectives after the Shift 3 3 0.5 2 4 0.753 0.158655 6.42 5 0.63 0.02275 43.45 6 0.53 0.00134996 740.76

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