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

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, per million to the right the mean + 4  there are per million to the right of the mean + 5  there are per million to the right of the mean + 6  there are 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  defective per million per million in one tail per million

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

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, 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

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

Working with 6  methodology you get 3.4 defectives per million How can this be, if the exact figure is ppm (or 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  ProcessCpk USL Right hand ppm defective 11 22 33 44 55 66 691, ,536 66,807 6, m+  m+2  m+3  1.5 m+4  m+5  m+6  Z score

Process Centred at m  Process Right hand ppm defective 11 22 33 44 55 66 691, ,536 66,807 6, Process Centred at m Cpk Right hand ppm defective Cp ,655 22,750 1,

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(x i -m) 2 E(L)=k  2

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, Loss Function 33 1.5   0.75  0.6  0.5  Standard Deviation 9k  k  2 1k  k  k  k  2

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, ,536 66,807 6, Loss Function 33 1.5   0.75  0.6  0.5  Standard Deviation 29.25k  k  k  k  k  k  2

Six Sigma Metric Cpk 11 22 33 44 55 66 Loss Function (Process Centred at m+1.5  ) 29.25k  k  k  k  k  k  2 Loss Function (Process Centred at m) 9k  k  2 1k  k  k  k  2 Cp

Six Sigma Metric Cpk 11 22 33 44 55 66 R H ppm defective (Process Centred at m+1.5  ) R H ppm defective (Process Centred at m) Cp ,655 22,750 1, , ,536 66,807 6,

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 Average Run Length 6.42

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

AVERAGE RUN LENGTH 6 Sigma process Probability to detect the change Average Run Length

Six Sigma Metric Standard Deviation 33 44 55 66 3    3 Probability of Defectives after the Shift Expected Number of samples to detect the Shift Average Run Length USL

Six Sigma Metric Standard Deviation 33 44 55 66 3    3 Probability of Defectives after the Shift Expected Number of samples to detect the Shift Average Run Length