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Regret Indices and Capability Quantification

second stage analyses again stress customer-orientation by ignoring the time-series order of observations, which is part of a producer's "insider information." The minima! technical reÄ…uirement of stage two analyses is to form realistic estimates for the marginal expected value (and variance) of loss under either current or ideaÅ‚ process operations. Specifically, we need to form a performance index, defined most simply as the ratio of observed loss to expected loss. Use of this sort of index scalÄ™ has at least two immediate payoffs: (1) it eliminates all dependence on the units in which loss was measured (dollars, francs, yen, etc.; see equation[12]), and (ii) it makes direct comparisons between diverse variables and processes theoretically possible and intuitively reasonable. We use a simple numerical example in our section on "Setting Quality Standards" to illustrate typical second stage analyses.

In stage three, the Cumulative Capability curve is defined to be the marginal cumulative distribution function (empirical or theoretical) of the performance index. When a process is "in contro!" statistically, there is no loss of information in ignoring time-series order. For all inferences that depend upon observed data only through the corresponding Iosses, the sample CC curve is then a complete, sufficient statistic. On the other hand, we also argue in our section on "Smoothing Regret Distributions" that it makes good sense to attempt to characterize the left-hand (Iow cost) taiÅ‚ of the marginal distribution of Iosses even when one's process is still "evolving" (has not yet stabilized.)

Other Major Topics We also describe an altemative formalism for rescaling of loss, known as "Poissonization of Regret," that defines equivalent nonconformances for variables data. The major advantages of this morÄ™ detailed form of analysis are outlined in our sections on "Smoothing Regret Distributions" and on forming "Composite Regret Indices" over time and/or across variables and processes. Other sections describe "An Environmental Monitoring Example" and discuss usage of CC curves in "Process Improvement Monitoring." Next, some new insights are given about minimum yields corresponding to values of Taguchi's Cp_m statistic and about the CC(1) summary statistic. The finaÅ‚ section contains an overall summary.

Regret Concepts

Suppose that a quantitative characteristic, X, can be measured either on each batch from a production process or at specific time intervals for a continuous process. Suppose that the intended target value for X is T, and let R(X) denote a function that measures long-term regret associated with potential deviations of X from T. Depending on one's point-of-view, regret is a cost-of-