Appendix 2: Product Upgrades Based on Minimum Expected Quality Loss 343
L n (x): Total quality loss per piece in the case of shape parameter n and
quality response x.
L n : Expected quality loss per piece in the case of shape parameter n and
quality response x.
According to the assumption A1 and Equations A2.1, A2.2, and A2.3, a
general quality loss function can be described as the following Equation
A2.4. Equation A2.4 covers all quality characteristics such as nominal-thebest, smaller-the-better, and larger-the-better.
General form
n
−n
L ( )
x = C + C x + C x
(A2.4)
n
b
s
l
After applying the assumption A2 into Equation A2.4, we can get Equations
A2.5 and A2.6 as follows. If the response of quality equals to the target value
(i.e., m), the total quality loss is to be zero (Equation A2.5) and the result of
differentiation for the response of quality having the target value (i.e., m) is
also to be zero as Equation A2.6.
n
−n
L (m) = C + C m + C m = 0
(A2.5)
n
b
s
l
n−1
n
′
− − 1
n
s
l
L (m) = nC x − nC x
= 0
(A2.6)
If we incorporate the specific value of n into Equations A2.5 and A2.6, we
obtain the general loss function as follows. If the value of n equals to 1, we
obtain the following results:
1
−1
L (m) = C + C m + C m = 0
(A2.7)
1
b
s
l
′
0
−2
n
s
l
L (m) = C m − C m = 0
(A2.8)
After solving Equations A2.7 and A2.8, we obtain the following results:
C = C m
2 , C = −2C m
l
s
b
s
If n equals to 2, we obtain the following results:
2
−2
L (m) = C + C m + C m = 0
(A2.9)
2
b
s
l
′
−3
2
s
l
L (m) = 2C m − 2C m = 0
(A2.10)