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Regret Indices and Capability Quantification

a power of 2 to emphasize that H is the X half-width [to regret half-height of K/2] in both [4] and [5].

Essential and Optional Properties of Regret Functions Regret functions have two essential properties. Regret is always non-negative

R(X) > 0 for -oo < X < +oo    [6]

And regret is always zero whenever X coincides with a target value

R(T) = O    [7]

A highly desirable third property is that R(X) be continuous and monotonically strictly increasing as the deviation of X from a single target value increases. This property is enjoyed by the ąuadratic, absolute value, logistic, and inverted normal examples discussed above, but is not absolutely required. Regrets of this special form will be called "2-to-l regrets" because the only information lost in converting an X ^ T into its R(X) is then simply information about which side of the target (left or right) that X came from.

It is also not essential that regret be computable from a single X measurement or that T be a fixed value. Specifically, regret could be defined in terms of the observed variance (or standard deviation) of observations within homogeneous subgroups or, say, in terms of how both the maximum and minimum observations within a homogeneous subgroup deviate from their target(s). In an inventory management setting, the X variable might measure stochastic demand from customers, while management adaptively Controls supply, which would then be a variable target, T, for demand.

Which Costs Are or Arę Not Represented by Regret? In quality control/assurance applications, R(X) is a proxy/surrogate for long-term economic impacts that are usually unknown and which could be tedious (if not impossible) to ąuantify exactly. R(X) doesn't address up-front costs, such as those for equipment modemization and statistical process control (SPC) training, designed to reduce process variation of X about T. In fact, those sorts of operational costs may be increased by quality/productivity programs that aim at "centering" a process (shifting the X distribution so that, say, its expected value converges to T) and/or reducing its variability. Producers should be willing (if not eager) to incur these up-front costs in hope of reducing potential long-range costs.