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Regret Indices and Capability Quantification

confidence, simultaneous lower and upper Iimits on CC(I); the region between these two bounds thus provides a central 90% confidence band for the unknown, true CC(I) curve.

When N is smali (less than 25, say), asymptotic bands tend to be too narrow. The exact critical values of Niederhausen (1981) could be used in these cases.

What Is the “Ideał” Form for a CC Curve? Assuming that you are using the most appropriate target value(s) and functional form of regret for the X characteristic of interest, your quality improvement objective should be to modify the process distribution of X so as to maximize the numerical value of the CC(I) curve at all regret index values, I. In other words, a CC(I) curve is ideally as HIGH as possible!

Because CC(I) is monotone non-decreasing function as I increases, the higher CC starts out at I = 0 of [10] and/or the faster the CC curve climbs as I increases, the higher is the implied capability of that process. Of course, CC(0)=1 will not be possible because, again, the area above a CC curve is the expected value of that regret index, which is usually at least approximately 1. For example, consider making comparisons among the three processes (A, B & C) whose CC(I) curves are super-imposed in Figurę 10 No one of these processes completely dominates the other two; all pairs of CC curves cross. However, although process C has the shortest right-hand-tail, process C also has lowest capability in the sense that it is clearly inferior to both A and B over the immensely important 0 < I < 1 rangę. In fact, process A would be my elear, personal choice for most capable. After all, A dominates B over the even wider rangę of at least 0 < I < 2. Key values from Figurę 10 are repeated in Table 3.

Notę, in particular, that the above "high capability" interpretation for CC curves is associated with achieving a skewed and/or long right-hand-tailed distribution of quality-costs relative to a fixed regret function. Given a fixed sample of X measurements on a process, it could be misleading to, instead, tailor one's choice of regret transformation to those data. For example, in Figurę 8, one could inerease apparent capability by using quadratic regret in a situation where logistic regret is most appropriate. Similarly, one could decrease apparent capability by using logistic regret in a situation where quadratic regret is most appropriate. Again, your quality improvement objective should be to modify the process distribution of X over time for one given, most appropriate regret function.