Development of the parametric tolerance modeling and optimization ...

where l*, and d 2 , are the optimal process mean and variance obtained
from the process parameter optimization phase. Although
the closed-from solution presented in Eq. (28) consists of the Lambert
W function component, its computation is very simple. A negative
value of d* is ignored because d is always greater than zero, so
the negative sign from Eq. (28) is removed. After computing d*, the
optimal LSL and USL can be respectively calculated from l* d*r*
and l* + d*r*.
5.1.3. Investigation of the second derivative and the conditions for
convexity
To verify the validity of d*, the second derivative is computed
and the conditions for obtaining the minimum value of E[TC2] are
investigated. The second derivative of E[TC2] with respect to d is
@ 2 E½TC2Š
@ 2 d
CR 2
¼ 2kd/ðdÞ þ 2r
k
ðl TÞ 2
d 2 r 2
: ð29Þ
d*obtained from Eq. (28) is the global minimum of E[TC2] if the value
of Eq. (29) at d* is greater than zero. Therefore, the following
conditions must be satisfied:
2kd/ðdÞ CR 2
þ 2r
k
ðl TÞ 2
d 2 r 2 > 0;
or
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CR þ 2kr
0 < d <
2 kðl TÞ 2
s
; ð30Þ
and
kr 2
CR þ 2kr 2 kðl TÞ 2
kr 2 > 0: ð31Þ
In many industrial situations, the validity of this solution is further
suggested by setting the value of CR to a very large number
compared to the values of l* T and r 2 (Phillips and Cho, 1998).
5.2. Case II
5.2.1. Process parameter optimization
Applying the condition l = T to Eq. (8) and then equating
oE[TC 1]/@r to zero, the closed-form solutions for the optimal process
mean l* and deviation r* are obtained as
l ¼ T; ð32Þ
and
r ¼ c3l þ c2
2k
or r ¼ c3T þ c2
: ð33Þ
2k
5.2.2. Tolerance optimization
Replacing l, USL, and LSL in Eqs. (11) and (13) with T, l + dr,
and l dr, respectively, and exploiting the properties defined in
Eq. (21), the expected quality loss E[L 2(y)] and the expected rejection
cost E[CR] are respectively presented as follows:
E½L2ðyÞŠ ¼ kr 2 ½2UðdÞ 2d/ðdÞ 1Š; ð34Þ
and
E½CRŠ ¼CR 1
Z d
/ðzÞdz ¼ 2CRf1 UðdÞg: ð35Þ
d
Substituting Eqs. (34), (35) and (16) into Eq. (17), the expected
total cost can be defined as
E½TC2Š ¼kr 2 f2UðdÞ 2d/ðdÞ 1gþ2CRf1 UðdÞg þ a0 þ 2a1dr :
ð36Þ
S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1733
The minimum total cost and d* can be obtained by minimizing
Eq. (36). The first derivative of E[TC 2] with respect to d is calculated
as follows:
@E½TC2Š
@d ¼ 2a1r 2CR/ðdÞþ2kd 2 r 2 /ðdÞ: ð37Þ
e 1
2 d2
Then, equating Eq. (37) to zero and substituting /(d) with
= ffiffiffiffiffiffi p
2p,
the result is given as
kr d 2 CR 1
e 2
kr 2 d2
¼ a1
p : ð38Þ
ffiffiffiffiffiffi
2p
d,
According to Proposition 2, substituting v, g1, g2, g3, and g4 with
kr*, CR=kr 2 pffiffiffiffiffiffi , 1/2, anda1 2p into Eq. (27), the closed-form
solution of d* is given by
d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Lambert W a1
C pffiffiffiffiffiffi
R
2kr 2p e
2
0
1
B
C CR
B
C
@ 2r k A þ
kr 2
v
u
;
t
ð39Þ
where l* and r 2 are the optimal process mean and variance obtained
from the process parameter optimization phase. After computing
d*, the optimal LSL and USL can be calculated from
l* d*r* and l* + d*r*, respectively.
5.2.3. Investigation of the second derivative and the conditions for
convexity
The validity of d* can be verified by the second derivative of
E[TC2] with respect to d. The closed-form solution presented in
Eq. (39) of d* is the global minimum if the following conditions
are satisfied:
@ 2 E½TC2Š
@ 2 d
¼ 2kd/ðdÞ CR
k þ 2r 2 ð1 d 2 Þ > 0;
or
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 < d < 1 þ CR
r
; ð40Þ
and
2kr 2
1 þ CR
> 0: ð41Þ
2kr 2
5.3. Case III
5.3.1. Process parameter optimization
After applying the conditions for this case to Eq. (8), the expected
total cost is the same as that of Case I. Thus, the optimal
process mean l* and deviation r* are determined by Eqs. (19)
and (20), respectively. For more details, please see Section 5.1.
5.3.2. Tolerance optimization
Replacing USL, and LSL in Eqs. (11) and (13) with T + dr, and
T dr, respectively, and exploiting the properties defined in Eq.
(21), the expected quality loss E[L2(y)] and the expected rejection
cost E[CR] are respectively presented as follows:
Z sþd
E½L2ðyÞŠ ¼ kðzr þ l TÞ 2 /ðzÞdz;
or
s d
n o
Uðs þ dÞ
k 2r ðl TÞþr 2
n o
ðs þ dÞ /ðs þ dÞ
k ðl TÞ 2 þ r 2
n o
Uðs dÞ
n o
/ðs dÞ; ð42Þ
E½L2ðyÞŠ ¼ k ðl TÞ 2 þ r 2
þ k 2r ðl TÞþr 2
ðs dÞ