Development of the parametric tolerance modeling and optimization ...

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A quality loss incurred due to the deviation from the target value of quality characteristic Y(T) is described as the quadratic function: k(y T) 2 , where k and T are 100 and 50, respectively. The quality loss coefficient k is a coefficient representing the magni- Fig. 6. Plots of E[TC1], E[L1], andE[CM1] with respect to r and l. Fig. 7. Plots of E[TC2], E[CR], E[L2], and E[CM2] with respect to d. Fig. 8. Plot of @E[TC 2]/@d with respect to d. S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1735 tude of the loss incurred by the deviation of y from the target value. As shown in Fig. 2, the relationship between the expected total cost, the quality loss, and the manufacturing cost is defined in Eq. (4) (i.e., E[TC 1]=E[L 1(y)] + E[CM 1]). If the loss coefficient k Fig. 9. Plot of @ 2 E[TC2]/@d 2 with respect to d. Fig. 10. Plots of E[TC 2], E[C R], E[L 2], and E[C M2] with respect to d. Fig. 11. Plot of @E[TC 2]/@dwith respect to d.
1736 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 increases, the expected quality loss E[L1(y)] and the expected total cost E[TC 1] also increase. When inspection is not implemented, the empirical model associated with manufacturing costs is described by the polynomial presented in Eq. (3) where its regression coefficients c0, c1, c2, and c 3 are 100, 6.1, 3.1, and 3.85, respectively. When inspection is implemented, the manufacturing cost (CM2) is defined as the polynomial model presented in Eq. (16) where a 0 and a 1 are 100 Fig. 12. Plot of @ 2 E[TC2]/@d 2 with respect to d. Fig. 13. Plots of the effect of r on E[TC1], E[L1], and E[CM1] when l = l*. Table 2 Effect of l on E[TC 1] when r = r*. l r E[L1(y)] E[CM1] E[TC1] 49.00 0.98 196.04 210.99 407.03 49.20 0.98 160.04 211.45 371.49 49.40 0.98 132.04 211.92 343.96 49.60 0.98 112.04 212.38 324.42 49.70 0.98 105.04 212.61 317.65 49.80 0.98 100.04 212.85 312.89 49.90 0.98 97.04 213.08 310.12 50.00 0.98 96.04 213.31 309.35 50.10 0.98 97.04 213.54 310.58 50.20 0.98 100.04 213.78 313.82 50.40 0.98 112.04 214.24 326.28 50.50 0.98 121.04 214.48 335.52 50.60 0.98 132.04 214.71 346.75 50.80 0.98 160.04 215.17 375.21 51.00 0.98 196.04 215.64 411.68 and 0.2, respectively. When the product performance determined by y falls outside the product specification limits, the rejection unit cost (CR) is set to an arbitrary value such as 100 in this numerical example. 6.1. Case I 6.1.1. Determining optimal settings of process parameters Using the closed-form solution given in Eqs. (19) and (20), l* and r* become 49.9883 and 0.9778, respectively. Fig. 2 plots Fig. 14. Plots of the effect of l on E[TC1], E[L1], and E[CM1] when r = r*. Table 3 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 1.2522 99.5993 21.0511 21.3398 141.9903 0.85 49.99 0.01 1.1790 99.5991 23.8395 21.1143 144.5529 0.90 49.99 0.01 1.1141 99.5989 26.5241 20.8129 146.9359 0.95 49.99 0.01 1.0561 99.5987 29.0913 20.4613 149.1513 1.00 49.99 0.01 1.0041 99.5984 31.5343 20.0792 151.2119 1.05 49.99 0.01 0.9571 99.5980 33.8506 19.6812 153.1298 1.10 49.99 0.01 0.9146 99.5976 36.0408 19.2786 154.9170 1.15 49.99 0.01 0.8759 99.5971 38.1075 18.8798 156.5845 1.20 49.99 0.01 0.8406 99.5965 40.0547 18.4912 158.1424 1.25 49.99 0.01 0.8084 99.5958 41.8868 18.1174 159.6001 1.30 49.99 0.01 0.7788 99.5950 43.6089 17.7623 160.9662 Fig. 15. Plots of the effect of r on E[TC 2], E[C R], E[L 2], E[C M2], and d* when l = l*.
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