Abstract

In the manufacturing process for the lens system of a mobile phone camera, various types of assembly and manufacturing tolerances, such as tilt and decenter, should be appropriately allocated. Because these tolerances affect manufacturing cost and the expected optical performance, it is necessary to choose a systematic design methodology for determining optimal tolerances. In order to determine the tolerances that minimize production cost while satisfying the reliability constraints on important optical performance indices, we propose a tolerance design procedure for a lens system. A tolerance analysis is carried out using Latin hypercube sampling for evaluating the expected optical performance. The tolerance optimization is carried out using a function-based sequential approximate optimization technique that can reduce the computational burden and smooth numerical noise occurring in the optimization process. Using the proposed design approach, the optimal production cost was decreased by 28.3% compared to the initial cost while satisfying all the constraints on the expected optical performance. We believe that the tolerance analysis and design procedure presented in this study can be applied to the tolerance optimization of other systems.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2010

M. Kehoe, “Tolerance assignment for minimizing manufacturing cost,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuF5.

R. Bates, “Performance and tolerance sensitivity optimization of highly aspheric miniature camera lenses,” Proc. SPIE 7793, 779302 (2010).
[CrossRef]

C.-C. Hsueh, P. D. Lin, and J. Sasian, “Worst-case-based methodology for tolerance analysis and tolerance allocation of optical systems,” Appl. Opt. 49, 6179–6188 (2010).
[CrossRef]

H. I. Choi, Y. Lee, D. H. Choi, and J. S. Maeng, “Design optimization of a viscous micropump with two rotating cylinders for maximizing efficiency,” Struct. Multidisc. Optim. 40, 537–548 (2010).
[CrossRef]

2009

PIAnO user’s manual, version 3.0, FRAMAX, Inc. (2009).

S. H. Lee and W. Chen, “A comparative study of uncertainty propagation methods for black-box-type problems,” Struct. Multidisc. Optim. 37, 239–253 (2009).
[CrossRef]

2008

D. J. Kim, J. H. Kim, and Y. B. Yoon, “A study of lens assembly deformation for mobile phone camera,” in Proceedings of the Annual Spring Conference of Korean Society of Mechanical Engineers (KSME, 2008), pp. 93–94.

G. Cassar, E. Vigier-Blanc, and T. Lépine, “Improved tolerancing for production yield anticipation of optical micro-modules for cameraphones,” Proc. SPIE 7100, 71000F (2008).
[CrossRef]

2007

CODE V 9.7 reference manual, Optical Research Associates (2007).

B. D. Youn, Z. Xi, L. J. Wells, and D. J. Gorsich, “Sensitivity-free approach for reliability-based robust design optimization,” in Proceedings of the 33rd Design Automation Conference, Presented at—2007 American Society of Mechanical Engineers International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007 (ASME, 2007), pp. 1309–1320.

S. K. Choi, R. V. Grandhi, and R. A. Canfield, Reliability-Based Structural Design (Springer, 2007).

2006

M. Isshiki, D. C. Sinclair, and S. Kaneko, “Lens design: global optimization of both performance and tolerance sensitivity,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper TuA5.

R. N. Youngworth, “21st century optical tolerancing: a look at the past and improvements for the future,” in International Optical Design, Technical Digest (CD) (Optical Society of America, 2006), paper MB3.

2005

S. H. Lee and B. M. Kwak, “Reliability based design optimization using response surface augmented moment method,” in Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization (ISSMO, 2005).

2004

S. Rahman and H. Xu, “A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics,” Prob. Eng. Mech. 19, 393–408 (2004).
[CrossRef]

H. Xu and S. Rahman, “A generalized dimension-reduction method for multidimensional integration in stochastic mechanics,” Int. J. Numer. Meth. Eng. 61, 1992–2019 (2004).
[CrossRef]

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[CrossRef]

2001

R. N. Youngworth and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
[CrossRef]

K. J. Hong, M. S. Kim, and D. H. Choi, “Efficient approximation method for constructing quadratic response surface model,” KSME Int. J. 15, 876–888 (2001).
[CrossRef]

S. Rahman and B. N. Rao, “A perturbation method for stochastic meshless analysis in elastostatics,” Int. J. Numer. Meth. Eng. 50, 1969–1991 (2001).
[CrossRef]

2000

R. N. Youngworth and B. D. Stone, “Cost-based tolerancing of optical systems,” Appl. Opt. 39, 4501–4512 (2000).
[CrossRef]

A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design (Wiley, 2000), Sec. 1.2.

A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods (SIAM, 2000).
[CrossRef]

1998

D. E. Huntington and C. S. Lyrintzis, “Improvements to and limitations of Latin hypercube sampling,” Prob. Eng. Mech. 13, 245–253 (1998).
[CrossRef]

1997

C. Y. Lin, W. H. Huang, M. C. Jeng, and J. L. Doong, “Study of an assembly tolerance allocation model based on Monte Carlo simulation,” J. Mater. Process. Technol. 70, 9–16 (1997).
[CrossRef]

1996

N. Alexandrov, “Robustness properties of a trust-region framework for managing approximations in engineering optimization,” in Proceedings of the 6th AIAA/NASA/USAF Symposium on Multidisciplinary Analysis and Optimization (AIAA, 1996), pp. 1056–1059.

R. M. Lewis, “A trust region framework for managing approximation models in engineering optimization,” in Proceedings of the 6th AIAA/NASA/USAF Symposium on Multidisciplinary Analysis and Optimization (AIAA, 1996), pp. 1053–1055.

1993

1988

F. Yamazaki, M. Shinozuka, and G. Dasgupta, “Neumann expansion for stochastic finite element analysis,” J. Eng. Mech. ASCE 114, 1335–1354 (1988).
[CrossRef]

1987

M. Stein, “Large sample properties of simulations using Latin hypercube sampling,” Technometrics 29, 143–151 (1987).
[CrossRef]

G. Adams, “Tolerancing of optical systems,” Ph.D. thesis (Imperial College, 1987).

R. Fletcher, Practical Methods of Optimization (Wiley, 1987).

1979

M. D. McKay, R. J. Beckman, and W. J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics 21, 239–245 (1979).
[CrossRef]

B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,” J. Eng. Mech. Div. ASCE 105, 661–676 (1979).

1975

M. J. D. Powell, “Convergence properties of a class of minimization algorithms,” in Nonlinear Programming(Academic, 1975).

1972

R. Fletcher, “An algorithm for solving linearly constrained optimization problem,” Math. Prog. 2, 133–165 (1972).
[CrossRef]

1970

1969

C. A. Cornell, “A probability-based structural code,” Am. Concrete Inst. J. 66, 974–985 (1969).

Adams, G.

G. Adams, “Tolerancing of optical systems,” Ph.D. thesis (Imperial College, 1987).

Alexandrov, N.

N. Alexandrov, “Robustness properties of a trust-region framework for managing approximations in engineering optimization,” in Proceedings of the 6th AIAA/NASA/USAF Symposium on Multidisciplinary Analysis and Optimization (AIAA, 1996), pp. 1056–1059.

Bates, R.

R. Bates, “Performance and tolerance sensitivity optimization of highly aspheric miniature camera lenses,” Proc. SPIE 7793, 779302 (2010).
[CrossRef]

Beckman, R. J.

M. D. McKay, R. J. Beckman, and W. J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics 21, 239–245 (1979).
[CrossRef]

Canfield, R. A.

S. K. Choi, R. V. Grandhi, and R. A. Canfield, Reliability-Based Structural Design (Springer, 2007).

Cassar, G.

G. Cassar, E. Vigier-Blanc, and T. Lépine, “Improved tolerancing for production yield anticipation of optical micro-modules for cameraphones,” Proc. SPIE 7100, 71000F (2008).
[CrossRef]

Chen, W.

S. H. Lee and W. Chen, “A comparative study of uncertainty propagation methods for black-box-type problems,” Struct. Multidisc. Optim. 37, 239–253 (2009).
[CrossRef]

Choi, D. H.

H. I. Choi, Y. Lee, D. H. Choi, and J. S. Maeng, “Design optimization of a viscous micropump with two rotating cylinders for maximizing efficiency,” Struct. Multidisc. Optim. 40, 537–548 (2010).
[CrossRef]

K. J. Hong, M. S. Kim, and D. H. Choi, “Efficient approximation method for constructing quadratic response surface model,” KSME Int. J. 15, 876–888 (2001).
[CrossRef]

Choi, H. I.

H. I. Choi, Y. Lee, D. H. Choi, and J. S. Maeng, “Design optimization of a viscous micropump with two rotating cylinders for maximizing efficiency,” Struct. Multidisc. Optim. 40, 537–548 (2010).
[CrossRef]

Choi, S. K.

S. K. Choi, R. V. Grandhi, and R. A. Canfield, Reliability-Based Structural Design (Springer, 2007).

Conn, A. R.

A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods (SIAM, 2000).
[CrossRef]

Conover, W. J.

M. D. McKay, R. J. Beckman, and W. J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics 21, 239–245 (1979).
[CrossRef]

Cornell, C. A.

C. A. Cornell, “A probability-based structural code,” Am. Concrete Inst. J. 66, 974–985 (1969).

Dasgupta, G.

F. Yamazaki, M. Shinozuka, and G. Dasgupta, “Neumann expansion for stochastic finite element analysis,” J. Eng. Mech. ASCE 114, 1335–1354 (1988).
[CrossRef]

Doong, J. L.

C. Y. Lin, W. H. Huang, M. C. Jeng, and J. L. Doong, “Study of an assembly tolerance allocation model based on Monte Carlo simulation,” J. Mater. Process. Technol. 70, 9–16 (1997).
[CrossRef]

Faaland, R. W.

Fiessler, B.

B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,” J. Eng. Mech. Div. ASCE 105, 661–676 (1979).

Fletcher, R.

R. Fletcher, Practical Methods of Optimization (Wiley, 1987).

R. Fletcher, “An algorithm for solving linearly constrained optimization problem,” Math. Prog. 2, 133–165 (1972).
[CrossRef]

Gardner, L.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[CrossRef]

Gorsich, D. J.

B. D. Youn, Z. Xi, L. J. Wells, and D. J. Gorsich, “Sensitivity-free approach for reliability-based robust design optimization,” in Proceedings of the 33rd Design Automation Conference, Presented at—2007 American Society of Mechanical Engineers International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007 (ASME, 2007), pp. 1309–1320.

Gould, N. I. M.

A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods (SIAM, 2000).
[CrossRef]

Grandhi, R. V.

S. K. Choi, R. V. Grandhi, and R. A. Canfield, Reliability-Based Structural Design (Springer, 2007).

Gregory, G. G.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[CrossRef]

Grey, D. S.

Grossman, L. W.

Haldar, A.

A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design (Wiley, 2000), Sec. 1.2.

Hong, K. J.

K. J. Hong, M. S. Kim, and D. H. Choi, “Efficient approximation method for constructing quadratic response surface model,” KSME Int. J. 15, 876–888 (2001).
[CrossRef]

Hsueh, C.-C.

Huang, W. H.

C. Y. Lin, W. H. Huang, M. C. Jeng, and J. L. Doong, “Study of an assembly tolerance allocation model based on Monte Carlo simulation,” J. Mater. Process. Technol. 70, 9–16 (1997).
[CrossRef]

Huntington, D. E.

D. E. Huntington and C. S. Lyrintzis, “Improvements to and limitations of Latin hypercube sampling,” Prob. Eng. Mech. 13, 245–253 (1998).
[CrossRef]

Isshiki, M.

M. Isshiki, D. C. Sinclair, and S. Kaneko, “Lens design: global optimization of both performance and tolerance sensitivity,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper TuA5.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[CrossRef]

Jeng, M. C.

C. Y. Lin, W. H. Huang, M. C. Jeng, and J. L. Doong, “Study of an assembly tolerance allocation model based on Monte Carlo simulation,” J. Mater. Process. Technol. 70, 9–16 (1997).
[CrossRef]

Kaneko, S.

M. Isshiki, D. C. Sinclair, and S. Kaneko, “Lens design: global optimization of both performance and tolerance sensitivity,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper TuA5.

Kehoe, M.

M. Kehoe, “Tolerance assignment for minimizing manufacturing cost,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuF5.

Kim, D. J.

D. J. Kim, J. H. Kim, and Y. B. Yoon, “A study of lens assembly deformation for mobile phone camera,” in Proceedings of the Annual Spring Conference of Korean Society of Mechanical Engineers (KSME, 2008), pp. 93–94.

Kim, J. H.

D. J. Kim, J. H. Kim, and Y. B. Yoon, “A study of lens assembly deformation for mobile phone camera,” in Proceedings of the Annual Spring Conference of Korean Society of Mechanical Engineers (KSME, 2008), pp. 93–94.

Kim, M. S.

K. J. Hong, M. S. Kim, and D. H. Choi, “Efficient approximation method for constructing quadratic response surface model,” KSME Int. J. 15, 876–888 (2001).
[CrossRef]

Kwak, B. M.

S. H. Lee and B. M. Kwak, “Reliability based design optimization using response surface augmented moment method,” in Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization (ISSMO, 2005).

Lee, S. H.

S. H. Lee and W. Chen, “A comparative study of uncertainty propagation methods for black-box-type problems,” Struct. Multidisc. Optim. 37, 239–253 (2009).
[CrossRef]

S. H. Lee and B. M. Kwak, “Reliability based design optimization using response surface augmented moment method,” in Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization (ISSMO, 2005).

Lee, Y.

H. I. Choi, Y. Lee, D. H. Choi, and J. S. Maeng, “Design optimization of a viscous micropump with two rotating cylinders for maximizing efficiency,” Struct. Multidisc. Optim. 40, 537–548 (2010).
[CrossRef]

Lépine, T.

G. Cassar, E. Vigier-Blanc, and T. Lépine, “Improved tolerancing for production yield anticipation of optical micro-modules for cameraphones,” Proc. SPIE 7100, 71000F (2008).
[CrossRef]

Lewis, R. M.

R. M. Lewis, “A trust region framework for managing approximation models in engineering optimization,” in Proceedings of the 6th AIAA/NASA/USAF Symposium on Multidisciplinary Analysis and Optimization (AIAA, 1996), pp. 1053–1055.

Lin, C. Y.

C. Y. Lin, W. H. Huang, M. C. Jeng, and J. L. Doong, “Study of an assembly tolerance allocation model based on Monte Carlo simulation,” J. Mater. Process. Technol. 70, 9–16 (1997).
[CrossRef]

Lin, P. D.

Lyrintzis, C. S.

D. E. Huntington and C. S. Lyrintzis, “Improvements to and limitations of Latin hypercube sampling,” Prob. Eng. Mech. 13, 245–253 (1998).
[CrossRef]

Maeng, J. S.

H. I. Choi, Y. Lee, D. H. Choi, and J. S. Maeng, “Design optimization of a viscous micropump with two rotating cylinders for maximizing efficiency,” Struct. Multidisc. Optim. 40, 537–548 (2010).
[CrossRef]

Mahadevan, S.

A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design (Wiley, 2000), Sec. 1.2.

McKay, M. D.

M. D. McKay, R. J. Beckman, and W. J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics 21, 239–245 (1979).
[CrossRef]

Neumann, H.-J.

B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,” J. Eng. Mech. Div. ASCE 105, 661–676 (1979).

Powell, M. J. D.

M. J. D. Powell, “Convergence properties of a class of minimization algorithms,” in Nonlinear Programming(Academic, 1975).

Rackwitz, R.

B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,” J. Eng. Mech. Div. ASCE 105, 661–676 (1979).

Rahman, S.

H. Xu and S. Rahman, “A generalized dimension-reduction method for multidimensional integration in stochastic mechanics,” Int. J. Numer. Meth. Eng. 61, 1992–2019 (2004).
[CrossRef]

S. Rahman and H. Xu, “A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics,” Prob. Eng. Mech. 19, 393–408 (2004).
[CrossRef]

S. Rahman and B. N. Rao, “A perturbation method for stochastic meshless analysis in elastostatics,” Int. J. Numer. Meth. Eng. 50, 1969–1991 (2001).
[CrossRef]

Rao, B. N.

S. Rahman and B. N. Rao, “A perturbation method for stochastic meshless analysis in elastostatics,” Int. J. Numer. Meth. Eng. 50, 1969–1991 (2001).
[CrossRef]

Sasian, J.

Shinozuka, M.

F. Yamazaki, M. Shinozuka, and G. Dasgupta, “Neumann expansion for stochastic finite element analysis,” J. Eng. Mech. ASCE 114, 1335–1354 (1988).
[CrossRef]

Sinclair, D. C.

M. Isshiki, D. C. Sinclair, and S. Kaneko, “Lens design: global optimization of both performance and tolerance sensitivity,” in International Optical Design, OSA Technical Digest (CD) (Optical Society of America, 2006), paper TuA5.

Stein, M.

M. Stein, “Large sample properties of simulations using Latin hypercube sampling,” Technometrics 29, 143–151 (1987).
[CrossRef]

Stone, B. D.

R. N. Youngworth and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
[CrossRef]

R. N. Youngworth and B. D. Stone, “Cost-based tolerancing of optical systems,” Appl. Opt. 39, 4501–4512 (2000).
[CrossRef]

Toint, P. L.

A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods (SIAM, 2000).
[CrossRef]

Vigier-Blanc, E.

G. Cassar, E. Vigier-Blanc, and T. Lépine, “Improved tolerancing for production yield anticipation of optical micro-modules for cameraphones,” Proc. SPIE 7100, 71000F (2008).
[CrossRef]

Wells, L. J.

B. D. Youn, Z. Xi, L. J. Wells, and D. J. Gorsich, “Sensitivity-free approach for reliability-based robust design optimization,” in Proceedings of the 33rd Design Automation Conference, Presented at—2007 American Society of Mechanical Engineers International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007 (ASME, 2007), pp. 1309–1320.

Xi, Z.

B. D. Youn, Z. Xi, L. J. Wells, and D. J. Gorsich, “Sensitivity-free approach for reliability-based robust design optimization,” in Proceedings of the 33rd Design Automation Conference, Presented at—2007 American Society of Mechanical Engineers International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007 (ASME, 2007), pp. 1309–1320.

Xu, H.

S. Rahman and H. Xu, “A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics,” Prob. Eng. Mech. 19, 393–408 (2004).
[CrossRef]

H. Xu and S. Rahman, “A generalized dimension-reduction method for multidimensional integration in stochastic mechanics,” Int. J. Numer. Meth. Eng. 61, 1992–2019 (2004).
[CrossRef]

Yamazaki, F.

F. Yamazaki, M. Shinozuka, and G. Dasgupta, “Neumann expansion for stochastic finite element analysis,” J. Eng. Mech. ASCE 114, 1335–1354 (1988).
[CrossRef]

Yoon, Y. B.

D. J. Kim, J. H. Kim, and Y. B. Yoon, “A study of lens assembly deformation for mobile phone camera,” in Proceedings of the Annual Spring Conference of Korean Society of Mechanical Engineers (KSME, 2008), pp. 93–94.

Youn, B. D.

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Figures (17)

Fig. 1
Fig. 1

Configuration of the lens system.

Fig. 2
Fig. 2

Lens and barrel assembly.

Fig. 3
Fig. 3

Types of surface tolerances.

Fig. 4
Fig. 4

Types of element tolerances.

Fig. 5
Fig. 5

Cost values at the lower limit, middle value, and upper limit for all the tolerance variables except image tilt. In the case of image tilt, the lower limit, middle value, and upper limit are 0, 20, and 40, respectively.

Fig. 6
Fig. 6

0.75 field points of a lens system.

Fig. 7
Fig. 7

PS result.

Fig. 8
Fig. 8

PS for thickness (TH) tolerances: (a)  R 1 and (b)  R 2 .

Fig. 9
Fig. 9

PS for surface decenter (SD) tolerances: (a)  R 1 and (b)  R 2 .

Fig. 10
Fig. 10

PS for surface tilt (ST) tolerances: (a)  R 1 and (b)  R 2 .

Fig. 11
Fig. 11

PS for element decenter (ED) tolerances: (a)  R 1 and (b)  R 2 .

Fig. 12
Fig. 12

PS for element tilt (ET) tolerances: (a)  R 1 and (b)  R 2 .

Fig. 13
Fig. 13

PS for image tilt (IT) tolerance: (a)  R 1 and (b)  R 2 .

Fig. 14
Fig. 14

Tolerance optimization procedure.

Fig. 15
Fig. 15

Tolerance optimization results for the (a) objective function and (b) probabilistic constraints.

Fig. 16
Fig. 16

Initial 2 n + 1 sampling points when n = 2 .

Fig. 17
Fig. 17

2 n new sampling points in the new trust region Γ k + 1 when n = 2 .

Tables (6)

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Table 1 Tolerances of the Lens System

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Table 2 Initial, Lower Bound, and Upper Bound Values of the Design Variables

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Table 3 Values of the Parameters in the Cost Function

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Table 4 Effects of Each Tolerance for R 1 in Descending Order of Importance

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Table 5 Effects of Each Tolerance for R 2 in Descending Order of Importance

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Table 6 Tolerance Optimization Results

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Cost = i = 1 6 ( t i · j = 1 N i ( a i j ( x i j ) 2 + b i j + c i j ) ) .
R 1 = P ( G 1 ( x ) G 1 U ) = G 1 ( x ) G 1 U f 1 ( x ) d x ,
R 2 = P ( G 2 ( x ) G 2 L ) = G 2 ( x ) G 2 L f 2 ( x ) d x ,
Find     x to minimize     Cost ( x ) subject to P ( G 1 ( x ) G 1 U ) R 1 L P ( G 2 ( x ) G 2 L ) R 2 L .
g k = [ a 11 a 21 a n 1 ] , D k = [ a 12 0 0 0 a 22 0 0 0 a n 2 ] ,
a i 2 = ( f i 2 f 0 ) ( x i 2 x 0 ) ( f i 1 f 0 ) ( x i 1 x 0 ) ( x i 2 x i 1 ) a i 1 = ( f i 1 f 0 ) ( x i 1 x 0 ) a i 2 ( x 0 + x i 1 ) .
G ˜ k = S k T [ G ˜ k 1 ( G ˜ k 1 p k ) ( G ˜ k 1 p k ) T p k T G ˜ k 1 p k + y k y k T y k T p k ] S k ,
B k = B k 1 ( B k 1 p k ) ( B k 1 p k ) T p k T B k 1 p k + y k y k T y k T p k .
f ˜ ( x ) = f ˜ ( x 0 ) + g 0 T ( x x 0 ) + 1 2 ( x x 0 ) T D 0 ( x x 0 ) ,
f ˜ ( x ) = f ( x k 1 ) + g k T ( x x k 1 ) + 1 2 ( x x k 1 ) T G ˜ k ( x x k 1 ) .
find     x k *     to minimize     f ˜ ( x )
subject to   h ˜ i ( x ) = 0 , i = 1 , , l
g ˜ j ( x ) 0 , j = 1 , , m
x Γ k ,
L ˜ 1 ( x ) = f ˜ ( x ) + [ i = 1 l | h ˜ i ( x ) | + j = 1 m max { 0 , g ˜ j ( x ) } ] ,
L 1 ( x ) = f ( x ) + [ i = 1 l | h i ( x ) | + j = 1 m max { 0 , g j ( x ) } ] .
Δ act k = L 1 ( x k ) L 1 ( x k * ) ,
Δ pred k = L 1 ( x k ) L ˜ 1 ( x k * ) .

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