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Regret Indices and Capability Quantification

pharmaceutical manufacturing facility provides an interesting application for Logistic regret, EÄ…uation [4]. The target value in all such cases is zero contaminant counts, but the half-width parameter, H, can vary depending upon where samples were collected —from product-contact surfaces, or air, or floors. Logistic regret is used here primarily to reduce sensitivity to outliers that can occur when, say, a floor sample is collected firom a visible footprint. In fact, the generaÅ‚ functional form of Logistic regret (steep toward the target, fiat in the taiÅ‚) is ideaÅ‚ when management wishes to focus on clean-up crew efifectiveness. Specifically, the vast majority of observed contaminant counts are already relatively smali, and the clean-up crew's primary objective is to make these smali counts even smaller. Elimination of outlying counts is an employee-awareness and/or manufacturing-capacity issue well beyond the control of clean-up crews.

My personal experience with contaminant count data is that they are frequently well approximated by a negative binomial distribution; the sample variances of counts are usually much larger than their means. One of the numerical examples I distribute with my personal Computer capability Ä…uantification software (see Appendix) contains 69 observed contaminant counts with mean 14.03, variance 238.94, and largest count 93. The Kolmogorov-Smirnov statistic for lack-of-fit of these counts to the negative binomial distribution (shape 1.507, mean 14.03, and variance 144.6) fitted via maximum likelihood (Johnson and Kotz, 1969) is 0.087, which is not significant at the 0.05 level. On the other hand, it is the stochastic distribution of regret (not that of the raw count data) that will be of primary interest for capability Ä…uantification.

The half-width parameter for Logistic regret (H in eÄ…uation (4J) will usually need to be greater than the observed mean count (14.03 here) because H represents the contamination level that is already one-half as undesirable as contamination could ever be. I used H = 20, and the resulting regret sample statistics were ER = 0.3383 and VR = 0.0326 (implied EE = 3.52). I rounded these values to ER = 0.333 and VR = 0.03, yielding a (gigantic) Poisson intensity of EE = 3.7 for each observation.

The Kolmogorov-Smimov statistic for lack-of-fit of observed Logistic regrets to this Poisson distribution is 0.121 (again, not significant at the 0.05 level). Figures 13 and 14 show a pair of graphical displays for this example drawn by my personal Computer software.

It is straight-forward to use cumulative capability curves in process improvement studies in the sense that there is a natural mechanism for updating these curves over time. Let us assume that ER and VR were first estimated from historical data, so that the CC(I) curve represents historical