- Volume 9 Issue 1
DOI QR Code
Optimization of Triple Response Systems by Using the Dual Response Approach and the Hooke-Jeeves Search Method
- Fan, Shu-Kai S. (Department of Industrial Engineering and Management Yuan Ze University) ;
- Huang, Chia-Fen (Department of Industrial Engineering and Management Yuan Ze University) ;
- Chang, Ko-Wei (Department of Industrial Engineering and Management Yuan Ze University) ;
- Chuang, Yu-Chiang (Department of Industrial Engineering and Management Yuan Ze University)
- 투고 : 2009.07.25
- 심사 : 2010.01.25
- 발행 : 2010.03.01
This paper presents an extended computing procedure for the global optimization of the triple response system (TRS) where the response functions are nonconvex (nonconcave) quadratics and the input factors satisfy a radial region of interest. The TRS arising from response surface modeling can be approximated using a nonlinear mathematical program involving one primary (objective) function and two secondary (constraints) functions. An optimization algorithm named triple response surface algorithm (TRSALG) is proposed to determine the global optimum for the nondegenerate TRS. In TRSALG, the Lagrange multipliers of target (secondary) functions are computed by using the Hooke-Jeeves search method, and the Lagrange multiplier of the radial constraint is located by using the trust region (TR) method at the same time. To ensure global optimality that can be attained by TRSALG, included is the means for detecting the degenerate case. In the field of numerical optimization, as the family of TR approach always exhibits excellent mathematical properties during optimization steps, thus the proposed algorithm can guarantee the global optimal solution where the optimality conditions are satisfied for the nondegenerate TRS. The computing procedure is illustrated in terms of examples found in the quality literature where the comparison results with a gradient-based method are used to calibrate TRSALG.
Triple Response System (TRS);Global Optimization;Nonlinear Programming (NLP);Trust Region (TR); Hooke-Jeeves' Search Method
- Bazaraa, M. S., Sherali, H. D., and Shetty, C. M. (1993), Nonlinear Programming: Theory and Algorithms, (3rd Ed.) A John Wiley and Sons, Inc., New York.
- Draper, N. R. (1963), 'Ridge Analysis' of response surfaces, Technometrics, 5(4), 469-479. https://doi.org/10.2307/1266023
- Del Castillo, E., Fan, S. K., and Semple, J. (1997), The computation of global optima in dual response systems, Journal of Quality Technology, 29, 347-353.
- Del Castillo, E., Fan, S-K., and Semple, J. (1999), Optimization of dual response system: a comprehensive procedure for degenerate and nondegenerate problems, European Journal of Operational Research, 112, 174-186. https://doi.org/10.1016/S0377-2217(97)00382-2
- Fan, S-K. (1996), Optimization of dual and multiple response processes, Ph.D. Dissertation, Department of Industrial Engineering University of Texas at Arlington, Arlington, TX.
- Fan, S. K. (2000), A generalized global optimization algorithm for dual response systems, Journal of Quality Technology, 32(4), 444-456.
- Fan, S. K. (2003), A different view of ridge analysis from numerical optimization, Engineering Optimization, 35(6), 627-647. https://doi.org/10.1080/03052150310001614846
- Hooke, R. and Jeeves, T. A. (1961), Direct Search Solution of Numerical and Statistical Problems, Journal of the Association Computer Machinery, 8, 212-229. https://doi.org/10.1145/321062.321069
- Lasdon, L. S., Fox, R. L., and Ratner, M. (1974), Nonlinear Optimization using the Generalized Reduced Gradient Method, Reveu Francaise d'Automatique et Recherche Operationnelle, 23, 73-104.
- Lasdon, L. S., Waren, A. D., Jain, A., and Ratner, M. (1978), Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming, ACM Transactions on Mathematical Software, 4(1), 34-50. https://doi.org/10.1145/355769.355773
- Luenberger, D. G. (1989), Linear and Nonlinear Programming, (2nd Ed.) Kluwer Academic Publishers, USA.
- Myers, R. H. and Carter, W. H. JR (1973), Response surface techniques for dual response systems, Technometrics, 15, 301-317. https://doi.org/10.2307/1266990
- Powell, M. J. D. (1998), Direct search algorithms for optimization calculations, Acta Numerica, 7, 287-336. https://doi.org/10.1017/S0962492900002841
- Sorensen, D. C. (1982), Newton's Method with a model trust region modification, SIAM J. Numer. Anal., 19(2), 409-426. https://doi.org/10.1137/0719026
- Semple, J. (1997), Optimality conditions and solution procedures for nondegenerate dual response systems, IIE Transactions, 29(9), 743-752.
- Shah, H. K., Montgomery, D. C., and Carlyle W. M. (2004), Response surface modeling and optimization in multiresponse experiments using seemingly unrelated regressions. Quality Engineering, 16(3), 387-397 https://doi.org/10.1081/QEN-120027941
- Vining, G. G. and Myers, R. H. (1990), Combining Taguchi and response surface philosophies: A dual response approach, Journal of Quality Technology, 22, 38-45.
- Gay, G. M. (1981), Computing optimal locally constrained step, SIAM J. Sci. Stat. Comput., 2(2), 186-197. https://doi.org/10.1137/0902016
- More, J. J., and Sorensen, D. C. (1983), Computing a trust region step, SIAM J. Sci. Stat. Comput., 4(3), 553-572. https://doi.org/10.1137/0904038