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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
1734 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 and E½CRŠ ¼CR Z s d /ðzÞdz þ 1 Z 1 sþd /ðzÞdz ¼ CRf1 Uðs þ dÞþUðs dÞg; ð43Þ where s =(T l*)/r*. Substituting Eqs. (42), (43) and (16) into Eq. (17), the expected total cost can be written as E½TC2Š ¼k ðl TÞ 2 þ r 2 n o Uðs þ dÞ k ðl TÞ 2 þ r 2 n o Uðs k 2r ðl TÞþr dÞ 2 n o ðs þ dÞ /ðs þ dÞ þ k 2r ðl TÞþr 2 n ðs o dÞ /ðs dÞ þ CRf1 Uðs þ dÞþUðs dÞg þ a0 þ 2a1dr : ð44Þ Even though the closed-form solution of d* for this case cannot be provided, the optimum expected total cost and d* can be found by minimizing Eq. (44). After d* is generated, the optimal LSL and USL can be computed from T d*r* and T + d*r*, respectively. 6. Numerical example To illustrate the application of the proposed models, an electronic chip manufacturing company experiencing high warranty costs and customer dissatisfaction associated with component failures in a prime product will be used. This company is considering a new process for manufacturing electronic chips. In view of the high Fig. 2. Plots of E[TC1], E[L1], andE[CM1] with respect to r and l. Table 1 Effect of r on E[TC 1] when l = l*. r l E[L1(y)] E[CM1] E[TC1] 0.50 49.99 25.01 307.16 332.17 0.60 49.99 36.01 287.60 323.61 0.70 49.99 49.01 268.05 317.06 0.80 49.99 64.01 248.49 312.50 0.90 49.99 81.01 228.93 309.94 1.00 49.99 100.01 209.38 309.39 1.10 49.99 121.01 189.82 310.83 1.20 49.99 144.01 170.27 314.28 1.30 49.99 169.01 150.71 319.72 1.40 49.99 196.01 131.15 327.16 1.50 49.99 225.01 111.60 336.61 1.60 49.99 256.01 92.04 348.05 1.70 49.99 289.01 72.48 361.49 1.80 49.99 324.01 52.93 376.94 1.90 49.99 361.01 33.37 394.38 2.00 49.99 400.01 13.82 413.83 costs associated with the failure of the components and the low inspection costs, the company follows a 100% inspection policy on a key quality characteristic Y which is normally distributed with mean l and standard deviation r. Fig. 3. Plots of E[TC2], E[CR], E[L2], and E[CM2] with respect to d. Fig. 4. Plot of @E[TC2]/@d with respect to d. Fig. 5. Plot of @ 2 E[TC 2]/@d 2 with respect to d.
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