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Obenchain

ąuadratic, inverted normal (H = 15ml), absolute value, and logistic (H = 15ml) regrets, respectively. Normally, one does not explore severaJ different forms of regret function on each set of data. Instead, one usually restricts attention to the single choice of R(X) function that seems to make the most sense from the point-of-view of the customer and/or regulator. But here our purpose is simply to show how different functional forms of regret can trigger different sorts of data-analytic challenges in setting ąuality standards.

Ouadratic Case The observed mean value of quadratic regret on the 31 samples was 74.06 ml squared. This value would be much lower if, say, the extreme observed fill-volume values of 51 and 98 ml were excluded from analysis, say, because they could be traced to "assignable causes." Anyway, the analysis given here is based upon setting the quadratic ER at 70 ml squared. For example, a fili value of 82 ml yields quadratic regret 7x7=49, and the corresponding performance index would be 1=49/70=0.70. Notę that the histogram of quadratic cost indices is highly skewed toward Iow costs, with 23 out of 31 indices being in the desirable "Iow" rangę of indices less than 1.0.

Logistic Case The observed mean value of logistic regret on the 31 samples is 0.23 units with the half-width set at H=15 ml. If a smaller logistic half-width were considered appropriate, then the logistic ER might increase to the point where it would become literally impossible to observe large performance indices. [For example, ER is approximately 0.40 at H=5 ml, and the resulting maximum possible index would then be only about 1/0.402.50.] Anyway, the analysis given here is based upon setting the logistic ER standard at 0.20 for H=15 ml. For example, a fili value of 82 ml yields logistic regret 7/(7+15)=0.32, and the corresponding performance index would be 1=0.32/0.20=1.59. Notę that the histogram of logistic indices shows no evidence of outliers, but only 17 out of 31 indices are in the desirable "Iow" rangę less than 1.0.

Inyerted Normal The observed mean value of inverted normal regret on the 31 samples was ER=0.15 units at H=15 ml. Here, a narrower half-width could be used without inflating ER above 0.25, and this would place much less emphasis on the extreme observed fill-volume values of 51 and 98 ml; currently, each of these extremes yields a mildly outlying performance index, I>5. For example, a fili value of 82 ml yields inverted normal regret 0.14, and the corresponding performance index would be 1=0.14/0.15=0.93. Notę that the histogram of inyerted normal indices is almost as highly skewed as is the quadratic regret