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Obenchain

poor-quality, or a measure of customer/regulatory dissatisfaction with products/services, or simply a formal decision-theoretic loss function. The best known example of a regret function is probably "loss to society," Taguchi and Wu (1979), which follows the familiar Ä…uadratic equation

R(X) = K(X-T)²    [1]

where K is a positive constant.

A discontinuous regret function is implied by the traditional approach where each indiyidual production unit either succeeds or fails to "conform to specifications," i.e., each individual X value gets converted into either a zero or a one, say. This regret function could be called "Attributes" or "Pass/Fail" or "Go/No go." But here it will be termed Goal-Posts regret and will be defined by

R(X) =0 if    T < X < T,

^v    12    _[2j

= 1 if X<T, or T₂<X

where Tj is the lower specification limit and T2 is the upper specification limit. This regret has an infinity of target-like values when Ti < T2; all values between T1 and T2 yield zero regret.

Of course, Goal-Posts regret can also be expressed in terms of a "central" target value [T = (Ti + T2)/2] and a "half-width" of zero regret [within ±H = (T2-Ti)/2 of the central value.] In this altemative parametrization, the central value, T, establishes "location" along the X-axis while the half-width, H, sets the "scalÄ™" along the X-axis.

Figures 1 and 2 illustrate Ä…uadratic and Goal-Posts regret functions; Figures 3, 4 and 5 illustrate three morÄ™ basie, symmetric-about-the-target fiinctional forms for regret. Specifically, Absolute Value regret is defined by

R(X) = K | X - T |    [3]

Logistic regret is defined by

R(X) = K | X - T | / [ | X - T | + H ]    [4]

and Inverted Normal regret is defined by

R(X) = K(l-2

-(X-T)²/H²

[5]