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Regret Indices and Capability Quantification

the regret function, R(X). Our analyses will be iterative in the sense that, although our choice of T or R(X) could change because of what we find in a later stage of our analyses, this sort of fundamental change invariably causes us to return to stage one and, essentially, start over almost from scratch!

Choice of target, T, is frequently immediate; T is usually the nominał specification for X. This target value could be the ideał dimension of a part, the zero-level of contamination, the label-claim (plus any overage required by industry-practice or regulation), etc.

Our choice of functional form for regret, R(X), could depend upon our engineering judgment, the opinions of subject matter experts and feedback from customers and regulators. The form of regret we adopt needs to be as realistic as possible in the sense that it truły mimics increases in customer/regulatory dissatisfaction as the X deviation from the target, T, increases. Thus, to get started, suppose that we have picked (at least tentatively) a target value, T, and a regret functional form, R(X).

Setting Quality Standards

In the critical, second stage of our analysis, we must examine the stochastic distribution of regret implied by the data at hand. Processes that are statistically "in contro!" are characterized by the property that all measurements madę on that process are, if not independent and identically distributed, at least interchangeable (or exchangeable), Barlow and Irony (1992). There is clearly no loss of information in ignoring time-series order in these cases. But the marginal distribution of regret usually tends to be highly skewed, asymmetric, and non-normal in shape. After all, regrets are always non-negative and yet regret distributions that are highly concentrated near zero are desirable!

Our primary CC stage-two task is to estimate a regret mean, ER, and a regret variance, VR, for a possibly highly-skewed marginal distribution. Thus our major concern when examining a sample of regret values can be whether we should "set aside" a portion of the available data before computing these summary statistics. For example, our sample may contain some very large "outlying" regret values, especially when regret is quadratic as in [1], These outliers may be due to special causes, see Shewhart (1931) and Gunter (199la). Our data may also span several distinct rational subsets, see AT&T (1956), Nelson (1988), and Gunter (199 lb). Within individual rational subsets, regrets measuring variability may be consistently lower (representing only common cause variation) than between subsets.

Specifically, our choice for expected regret, ER, defines our primary ąuality standard. This standard ER is critical because it will be the