Argon Chen Vic Chen Chris Hsu Graduate Institute Of .

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Ridge Search for Statistical Multi-objectiveOptimization and Application toIC Geometric DesignArgon Chen Vic Chen Chris HsuGraduate Institute of Industrial EngineeringNational Taiwan UniversityEffects of Rounded Corners on E-TestsRegion perbound0.4E10.050.15W0.10.321

Empirical Model of Effects on E-TestsE-test f (H1, E1, W ) εyˆ b0 bx x T BxResponse Surfacex1FactorsxkxKy1 f 1(x1, x2, , xk ) ε1y2 f 2(x1, x2, , xk ) ε2 yi f (x1, x2, , xk ) ειE-tests y1yiyIεOptimization Formulation and Results Layout factor xk window:Lk xk Uk E-test output with target Ti :min wi ( yi Ti )2 wi (bi0 bi x xT Bi x Ti )xki2i Minimizing sensitivity of yi to xk changes:min (xk(i ,k ) y i 0) 2 x k E-test output with spec. windows :L j y j U j L j b j 0 b j x xT B j x U j2

11120.1110.080.3XX10.280.29 0.11080.11060.1104XX20.260.1Inclined Trough0.120.29820.11026 66066060 6067-0 0-0 -00 0e-0 06 6 -0-0 0 -0 07e-0-0 06066 7e5e2e45 e-0-0 08 3e-0 0 7 075ee8e 0 0 e-061 7e-0 06 e-0-0002026103856 0 61.662 80 51e.39 6 .151 -0 0760 202 3e- 7 82.989768 25ee-00 6675523ee2-. 2. 2.04 e-01.15.48 e-0-00 061 9 3e9.87. 4 5805.75240 8 07.662.714 e-0-20. .3 .318e 1.061.92.763429 7e 2 21.323372 .500712003e67357e-0 0 64.9 -0 07 4e-0 0477e -0 0e9.0 791 5e2 8 902-0 01.8 09 6.11 3.742e1.8 -0 000253e 69e- -0 07606 07 8.2 0 07 7 0 076727 -00 6 -0 0e2. 4 645 4e-0 -01. 745 ee-0-008063ee- 0 e1 5 6 2e2 8 e-0 7 81.1. 662 80 51e. 39 0 6 . 15 e-.08070. 40 0 3 . 758751.15. 48 e-0 0 061 9 3 96. 5616 8e- 1.0071.3233-03e107735-0007e4.9 -0 07 4e470e9.02 8 . 29130707-04.172ee-00-02579864e8.2 0 07 7 07002. 4 645ee- 0 e-0911.2 8 e-0 7 82240 3 58.7. . 580 5. 767800-0 0e1e53290779378.2-0 04. -0 0 4e-04ee 0452 8 291.611 3.4.50.160.2984070.2996X XXXXXXXXXXXXX0.29862.4679e-060.290.080.3Zigzagging clined Trough0.14X8.229 1e-0087-0 03e3507-0 074.95 4ee-01.6467 92. 407e-090 473. 2-0 0077 2ee-00 0 25 0712 88e- 8. 2 7e-04.1 -0 07 070247-0e7.49.0 070635 3 7 8e4. 9 5.756-0 06 e-02e -0 1 5 06 000770 3e15 -0 ee-09.81. 069 1.3318.e31680 36. 51.26-0 0 63e098 0 06-0 0 61.3 5e-2 8e -0680 56 45 e 06 6 0 01.4 1. .63e-0-0107e127 9 5.e89 0 6 061.17. 80 14 e-20e-07 61.29. 050.2988XXX7-0 007 2ee-0. 225 0702 8 8 7 7e-07. 449. 0 0706e-0 0 065e-070 2 e- 519. 8 . 069 31.110.29922.4679e0071X. 645 4e-00774e-0 03.29078e-0 04.112075 3e-0 e-0 074.93785.75-0 070 3e6.583Obj valuex 10X-3X0.299x23Obj valueProblem of Gradient Search Method(Ex. GRG)6x 10-3542100.16Zigzag of GRG Method3

Ridge AnalysisQuadratic Response SurfaceMinimizeRidge PathOriginsubject toyˆ b0 bT x xT BxxT x R21 xˆ s (B µI ) 1 b2Generalized Ridge Search MethodHessian Matrix for Quadratic ApproximationGk µI ) 1 β k( xs x k 22( )f xs4 x1s( )2x 1sx s3 f ( xs ) f f ( )µ (γ 1) µ (γ ) α γ x s2 x s3 x s4R4R3R1R2µxk04

Quartic FunctionQuadratic functionQuadratic of quadraticTrough in Quartic Function400350300250200150100500Trough Ring0.080.070.06Trough0.050.040.03Quadratic of quadraticX10.350.340.320.330.310.30.29X25

Constrained Ridge Search Method65 3 25.65 472273627 62. 6.1379 286 2.1801961 56631593 1 08.110 00745 39 44.61.50 70.38330.3614 20 998.0972734 881 17. 5 54.00.562216 634 81127.0541833067 62.56 35.16 1472 2737285 89.150 71.2816928.6379 2 .65656538 36193 145 0833 81.039.141400 191.600728 17.4582022 594.0727916 9011 27.0541 .5634 88563.54487335 03301690.56348Line Searchd3035 033 0Maximum300.0230Decreasing0357348.5 4391422 6 90454.654 .5610800.0 6372 49.1 733678.679 8108127632.379 5650 70 1.091 8 172514 2 35. 1.584. 67212615628693205392 10.01303561805427.110.26920.5817280.2890.322 54.07270.324487335 030?0.3456 3.533 0Minimum30Increasing3303300.4? f ( x k ) T d 01156273.5.05344 5 0418783303033 00.030.04300.050.060.07Design for Manufacturability (DFM) Case (i)Region ofroundingphenomenonDesign factors on geometric layout H1 Æ0.4 ; E1 Æ0.05 changes in E1 and H1 should not havemuch influence on the 50.15W0.10.3Specification windowResponseDesiredTarget 8-0.2789B10.470.42120.5148C1240217.3421265.64036

Design for Manufacturability (DFM) Case (ii)Stopping Criterion: bias 10-6 or 700 iterations3 Variables 26 constraintsGRR Zoutendijk23 initial solutions8 feasible initial solutionsrchl SeaLocaBestObjective ValueAverageObjective ValueAverage StepAverageComputing TimeGRG Zoutendijk1.559253 1072.129230 107619.132.44 secondsGRR Zoutendijk1.559223 1071.559223 10734.630.51 seconds10710712Less than 1 secondLingoarchal SeGlob1.559223 1.559223 Objective ValueStepComputing TimeGRG Zoutendijk1.559253 107495319.69 secondsGRR Zoutendijk1.559223 1072774.30 seconds10784922 secondsLingo1.559223 7

Argon Chen Vic Chen Chris Hsu Graduate Institute of Industrial Engineering National Taiwan University 2 Effects of Rounded Corners on E-Tests rounded corner OPC Intended Polygon Design Resulting Print H1 E1 W Region of rounding effect W 0.1 0.3 E1 0.05 0.15 H1 0 0.4 Upper bound Design Lower Factors

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