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solving for K:
A
K
=
(18.2)
y ) 2
(
In the Taguchi tolerance design method, the quality loss coefficient K can be
determined based on losses in monetary terms by falling outside the customer toler-
ance limits (design range) instead of the specification limits usually used in process
capability studies, for example, or the producer limits . The specification limits most
often are associated with the design parameters. Customer tolerance limits are used to
estimate the loss from customer perspective or the quality loss to society as proposed
by Taguchi. Usually, the customer tolerance is wider than manufacturer tolerance. In
this chapter, we will side with the design range limits terminology. Deviation from
this practice will be noted where needed.
Let f(y) be the probability density function (pdf) of the y , then via the expectation
operator, E , we have the following:
K σ
T y ) 2
y
E [ L ( y
,
T )]
=
+
(
µ y
(18.3)
Equation (18.3) is fundamental. Quality loss has two ingredients: loss incurred
as a result of variability
σ
y
and loss incurred as a result of mean deviation from
T y ) 2 . Usually the second term is minimized by adjusting the mean of
the critical few design parameters —the affecting x's.
The derivation in (18.3) suits the nominal-is-best classification. Other quality loss
function mathematical forms may be found in El-Haik (2005). The following forms
of loss function were borrowed from their esteemed paper.
target (
µ y
18.4.1
Larger-the-Better Loss Function
For functions like “increase yield” ( y
=
yield), we would like a very large target,
ideally T y →∞
. The requirement (output y ) is bounded by a lower functional spec-
ifications limit y l . The loss function then is given by
K
y 2
L ( y
,
T y )
=
where
y
y l
(18.4)
µ y be the average y numerical value of the software range (i.e., the average
around which performance delivery is expected). Then by Taylor series expansion
around y
Let
= µ y , we have
K 1
µ
E L ( y
T y ) =
3
µ
2
y
,
y +
y σ
(18.5)
 
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