00392 °9c4879e21ccf146f13d3533b34bd5d

396

Obenchain

States on page 17 that PCI = (UTL - LTL) / 6a ^His sometimes called Cp/' where UTL and LTL are Upper and Lower Tolerance Limits.

The generaÅ‚ expression [Chan, Cheng, and Spiring (1988), Taguchi, Elsayed, and Hsiang (1989), Boyles (1991)] for the Cp_m index when the process may be off-target and regret is quadratic with K = 1 in Equation [1] is

[24]

, UTL-LTL

6-VIr where ER = a² + (p-T)², as in Equation [9], without assuming normality; Johnson (1992) stresses this squared-error-loss interpretation of Cpm- Boyles (1991) discusses several specific advantages of Cp_m over both Cp and Cpk. Authors who have discussed confidence bounds for Cp, Cpk, and Cp_m indices include Bissell (1990), Zhang, Stenback and Wardrop (1990), Boyles (1991), Kushler and Hurley (1992), Franklin and Wasserman (1992) and Guirguis and Rodriguez (1992).

Taguchi's early writings about Cp_m (1979) used the symbol s' to denote the "standard deviation from the target," VeR , rather than the usual standard deviation (from the mean, p). Of course, >/ER does simplify to the usual s and Cpm = Cp in special cases where p = T, i.e. when the process is on-target-on-average. In these special cases, [24] does have the following highly intuitive interpretation: Cp is then the ratio of the tolerance rangÄ™ to the width of a three sigma control chart.

Process yields (fraction of production conforming to tolerances) "usually" increase as Cp_m increases, at least when the target is "near" the tolerance mid-point. Furthermore, Boyles (1991) shows that large numerical values of Cp_m can only occur when the process mean is near its target;

| p - T| cannot exceed (UTL-LTL)/(6 • Cpm).

Under the common assumption that p = T = (LTL+UTL)/2, a distribution-free lower limit can be placed on process yield as a function of Cp_m=Cp. In these special cases, we have

Prob( LTL < x < UTL ) = Prob[|x - p|< (UTL - LTL) / 2]

= Prob(|x-p|<3 C_p a]

= l-Prob[|x-p|> 3 C_p-a]

[25]