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E[TC1] versus l and r, where the minimum value of E[TC1] obtained at l* and r* is 309.3380. The sensitivity analysis of E[TC1] to the variation in process standard deviation when the process mean is fixed at l* is presented in Table 1 and Fig. 13. A similar analysis is conducted when the process standard deviation is fixed at r* and it is shown in Table 2 and Fig. 14. 6.1.2. Determining optimal tolerance Using the closed-form solution defined in Eq. (28), d* becomes 1.0269. This optimal value generates the minimum E[TC 2], optimal LSL, and optimal USL at 150.3168, 48.9842, and 50.9924, respec- Table 4 Effect of jl Tj on d* when r = r*. l jl Tj r d* E[CM2] E[CR] E[L2(y)] E[TC2] 49.0 1.00 0.98 0.0685 99.9731 94.5356 5.4726 199.9813 49.1 0.90 0.98 0.4516 99.8230 65.1579 30.4354 195.4162 49.2 0.80 0.98 0.6182 99.7577 53.6425 35.0573 188.4574 49.3 0.70 0.98 0.7345 99.7121 46.2616 34.9641 180.9377 49.4 0.60 0.98 0.8222 99.6777 41.0981 32.8406 173.6164 49.5 0.50 0.98 0.8896 99.6513 37.3676 29.9179 166.9368 49.6 0.40 0.98 0.9412 99.6310 34.6598 26.8931 161.1839 49.7 0.30 0.98 0.9795 99.6160 32.7348 24.1974 156.5482 49.8 0.20 0.98 1.0059 99.6057 31.4458 22.1055 153.1570 49.9 0.10 0.98 1.0215 99.5996 30.7041 20.7882 151.0919 50.0 0.00 0.98 1.0266 99.5976 30.4619 20.3391 150.3987 Fig. 16. Plots of the effect of jl Tj on E[TC2], E[CR], E[L2], E[CM2], and d* when r = r*. Table 5 Effect of r on E[TC 1] when l = l*. r l E[L1(y)] E[CM1] E[TC1] 0.5 50.00 25.00 307.20 332.20 0.6 50.00 36.00 287.64 323.64 0.7 50.00 49.00 268.08 317.08 0.8 50.00 64.00 248.52 312.52 0.9 50.00 81.00 228.96 309.96 1.0 50.00 100.00 209.40 309.40 1.1 50.00 121.00 189.84 310.84 1.2 50.00 144.00 170.28 314.28 1.3 50.00 169.00 150.72 319.72 1.4 50.00 196.00 131.16 327.16 1.5 50.00 225.00 111.60 336.60 1.6 50.00 256.00 92.04 348.04 1.7 50.00 289.00 72.48 361.48 1.8 50.00 324.00 52.92 376.92 1.9 50.00 361.00 33.36 394.36 2.0 50.00 400.00 13.80 413.80 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1737 tively. Plots of E[TC2], E[CM2], E[CR] and E[L2] with respect to d are presented in Fig. 3. Figs. 4 and 5 show that the first derivative of E[TC2] with respect to d at d* is zero and the second derivative of E[TC 2] with respect to d at d* is greater than zero, respectively. The latter figure also validates the conditions for convexity presented in Eqs. (30) and (31). Table 6 Effect of l on E[TC 1] when r = r*. l r E[L1(y)] E[CM1] E[TC1] 49.0 0.98 96.04 210.99 307.03 49.2 0.98 96.04 211.45 307.49 49.4 0.98 96.04 211.92 307.96 49.6 0.98 96.04 212.38 308.42 49.7 0.98 96.04 212.61 308.65 49.8 0.98 96.04 212.85 308.89 49.9 0.98 96.04 213.08 309.12 50.0 0.98 96.04 213.31 309.35 50.1 0.98 96.04 213.54 309.58 50.2 0.98 96.04 213.78 309.82 50.3 0.98 96.04 214.01 310.05 50.4 0.98 96.04 214.24 310.28 50.5 0.98 96.04 214.48 310.52 50.6 0.98 96.04 214.71 310.75 50.8 0.98 96.04 215.17 311.21 51.0 0.98 96.04 215.64 311.68 Fig. 17. Plots of the effect of r on E[TC 1], E[L 1], and E[C M1] when l = l*. Fig. 18. Plots of the effect of l on E[TC 1], E[L 1], and E[C M1] when r = r*.
1738 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 This is a solid proof that E[TC2] is a convex function and d* is the global minimum because its value is between 0 and 1.7452. The sensitivity analysis of d* to the change in r can be seen in Table 3 or Fig. 15. It can be observed that d* gradually decreases as r increases. Additionally, E[CM2] depends on d* and r and its value is the lowest when r is 1.30. Similarly, the sensitivity analysis of the effect of the process bias (or jl Tj)ond* is shown in Table 4 or Fig. 16. It is shown that, as the process bias increases, d* decreases while E[CR] increases. For the 100% inspection of Y, E[L2] is more affected by the process bias than by r, whereas E[C R]is sensitive to both. Table 7 Effect of r on d* when l = l*. r l l Tj d* E[CM2] E[CR] E[L2(y)] E[TC2] 0.80 50.00 0.00 1.2555 99.5982 20.9297 21.4541 141.9821 0.85 50.00 0.00 1.1815 99.5983 23.7405 21.2063 144.5451 0.90 50.00 0.00 1.1158 99.5983 26.4520 20.8781 146.9284 0.95 50.00 0.00 1.0570 99.5983 29.0509 20.4949 149.1442 1.00 50.00 0.00 1.0041 99.5983 31.5310 20.0756 151.2050 1.05 50.00 0.00 0.9563 99.5983 33.8904 19.6344 153.1232 1.10 50.00 0.00 0.9129 99.5983 36.1303 19.1818 154.9105 1.15 50.00 0.00 0.8732 99.5983 38.2539 18.7256 156.5778 1.20 50.00 0.00 0.8369 99.5983 40.2658 18.2713 158.1354 1.25 50.00 0.00 0.8035 99.5983 42.1712 17.8230 159.5925 1.30 50.00 0.00 0.7726 99.5982 43.9758 17.3836 160.9576 Fig. 19. Plots of the effect of r on E[TC 2], E[C R], E[L 2], E[C M2], and d* when l = l*. Table 8 Effect of r on d* when l = l*. r l jl Tj d* E[CM2] E[CR] E[L2(y)] E[TC2] 0.80 49.99 0.01 1.2555 99.5982 20.9346 21.4538 141.9866 0.85 49.99 0.01 1.1815 99.5983 23.7450 21.2058 144.5491 0.90 49.99 0.01 1.1158 99.5983 26.4560 20.8776 146.9319 0.95 49.99 0.01 1.0570 99.5983 29.0546 20.4944 149.1473 1.00 49.99 0.01 1.0041 99.5983 31.5344 20.0750 151.2077 1.05 49.99 0.01 0.9563 99.5983 33.8935 19.6338 153.1256 1.10 49.99 0.01 0.9129 99.5983 36.1330 19.1813 154.9126 1.15 49.99 0.01 0.8732 99.5983 38.2564 18.7250 156.5797 1.20 49.99 0.01 0.8369 99.5983 40.2680 18.2708 158.1371 1.25 49.99 0.01 0.8035 99.5983 42.1732 17.8226 159.5941 1.30 49.99 0.01 0.7726 99.5982 43.9776 17.3831 160.9590 6.2. Case II 6.2.1. Determining optimal settings of process parameters Similarly to case I, the minimum value of E[TC 1] obtained at the optimal process mean and standard deviation (which are determined by Eqs. (32) and (33), i.e., l* = 50.0000 and r* = 0.9780) is 309.3516 as illustrated in Fig. 6. In addition, the results of the sensitivity analysis of E[TC 1] are given in Tables 5 and 6, and Figs. 17 and 18. 6.2.2. Determining optimal tolerance Following the closed-form solution presented in Eq. (39), d* becomes 1.0267. This optimal value generates the minimum E[TC2], the optimal LSL, and the optimal USL at 150.3166, 48.9959, and 51.0041, respectively. Plots of E[TC2], E[CM2], E[CR] and E[L2(y)] with respect to d are presented in Fig. 7. Fig. 8 shows that the first derivative of E[TC2] with respect to d becomes zero when d is at the optimum point. The conditions of convexity for E[TC2] discussed in Section 5.2 are presented in Fig. 9, which shows that when d is at the optimum point (d*), the value of @ 2 E[TC2]/@d 2 is greater than zero. It also shows the range of d, which can be derived from Eq. (40). The result is the range between 0 and 1.2340, which contains the value of d*. The effect of r on d* can be seen in Table 7 or Fig. 19. It can be observed that d* decreases as r increases, and E[CR] is inversely proportional to d*. For this case, the effect of the process bias (or jl Tj) ond* is not provided because no l in the closed-form solution of d* defined in Eq. (39). It can be concluded that l has no effect to d*. For the 100% inspection of Y, to decrease the total cost, E[L2] and r should be controlled because they are directly proportional to the total cost. Fig. 20. Plots of the effect of r on E[TC2], E[CR], E[L2], E[CM2], and d* when l = l*. Table 9 Effect of jl Tj on d* when r = r*. l jl Tj r d* E[CM2] E[CR] E[L2(y)] E[TC2] 49.0 1.00 0.98 1.0272 99.5974 51.7607 16.0869 167.4449 49.1 0.90 0.98 1.0271 99.5974 48.2578 16.8773 164.7325 49.2 0.80 0.98 1.0270 99.5974 44.9205 17.5967 162.1146 49.3 0.70 0.98 1.0270 99.5974 41.8078 18.2389 159.6441 49.4 0.60 0.98 1.0269 99.5974 38.9766 18.7994 157.3734 49.5 0.50 0.98 1.0269 99.5974 36.4801 19.2753 155.3528 49.6 0.40 0.98 1.0269 99.5974 34.3663 19.6651 153.6289 49.7 0.30 0.98 1.0269 99.5974 32.6769 19.9680 152.2423 49.8 0.20 0.98 1.0269 99.5974 31.4452 20.1840 151.2267 49.9 0.10 0.98 1.0269 99.5974 30.6961 20.3135 150.6070 50.0 0.00 0.98 1.0269 99.5974 30.4448 20.3565 150.3987
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