Advanced SearchSearch Tips
Multi-Optimal Designs for Second-Order Response Surface Models
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
Multi-Optimal Designs for Second-Order Response Surface Models
Park, You-Jin;
  PDF(new window)
A conventional single design optimality criterion has been used to select an efficient experimental design. But, since an experimental design is constructed with respect to an optimality criterion pre specified by investigators, an experimental design obtained from one optimality criterion which is superior to other designs may perform poorly when the design is evaluated by another optimality criterion. In other words, none of these is entirely satisfactory and even there is no guarantee that a design which is constructed from using a certain design optimality criterion is also optimal to the other design optimality criteria. Thus, it is necessary to develop certain special types of experimental designs that satisfy multiple design optimality criteria simultaneously because these multi-optimal designs (MODs) reflect the needs of the experimenters more adequately. In this article, we present a heuristic approach to construct second-order response surface designs which are more flexible and potentially very useful than the designs generated from a single design optimality criterion in many real experimental situations when several competing design optimality criteria are of interest. In this paper, over cuboidal design region for variables, we construct multi-optimal designs (MODs) that might moderately satisfy two famous alphabetic design optimality criteria, G- and IV-optimality criteria using a GA which considers a certain amount of randomness. The minimum, average and maximum scaled prediction variances for the generated response surface designs are provided. Based on the average and maximum scaled prediction variances for k = 3, 4 and 5 design variables, the MODs from a genetic algorithm (GA) have better statistical property than does the theoretically optimal designs and the MODs are more flexible and useful than single-criterion optimal designs.
Multi-optimal designs;G-efficient design;IV-efficient design;genetic algorithms;cuboidal regions;
 Cited by
A Case Study to Select an Optimal Split-Plot Design for a Mixture-Process Experiment Based on Multiple Objectives, Quality Engineering, 2014, 26, 4, 424  crossref(new windwow)
Atwood, C. L. (1969). Optimal and efficient designs of experiments, Annals of Mathematical Statistics, 40, 1570-1602 crossref(new window)

Borkowski, J. J. (1995a). Finding maximum G-criterion values for central composite designs on the hypercube, Communications in Statistics: Theory and Methods, 24, 2041-2058 crossref(new window)

Borkowski, J. J. (1995b). Minimum, maximum, and average spherical prediction variances for central composite and Box-Behnken Designs, Communications in Statistics: Theory and Methods, 24, 2581-2600 crossref(new window)

Borkowski, J. J. (1995c). Spherical prediction-variance properties of central composite and Box-Behnken designs, Technometrics, 37, 399-410 crossref(new window)

Borkowski, J. J. (2003). Using a generic algorithm to generate small exact response surface designs, Journal of Probability and Statistical Science, 1, 65-88

Borkowski, J. J. and Valeroso, E. S. (2001). Comparison of design optimality criteria of reduced models for response surface designs in the hypercube, Technometrics, 43, 468-477 crossref(new window)

Box, G. E. P. and Behnken, D. W. (1960). Some new three-level designs for the study of quantitative variables, Technometrics, 30, 1-40 crossref(new window)

Box, G. E. P. and Draper, N. R. (1987). Empirical Model-Building and Response Surfaces, John Wiley & Sons, New York

Box, G.E.P. and Wilson, K.B. (1951). On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, 13, 1-45

Clyde, M. and Chaloner, K. (1996). The equivalence of constrained and weighted designs in multiple objective designs problems, Journal of the American Statistical Association, 91, 1236-1244 crossref(new window)

Cook, R. D. and Nachtsheim, C. J. (1980). A comparison of algorithms for constructing exact D-optimal designs, Technometrics, 3, 315-324 crossref(new window)

Cook, R. D. and Nachtsheim, C. J. (1982). Model robust, linear-optimal designs, Technometrics, 24, 49-54 crossref(new window)

Cook, R. D. and Wong, W. K. (1994). On the equivalence of constrained and compound optimal designs, Journal of American Statistical Association, 89, 687-692 crossref(new window)

Evans, G. W. (1984). An overview of techniques for solving multi-objective mathematical programs, Management Science, 30, 1268-1282 crossref(new window)

Fedorov, V. V. (1972). Theory of Optimal Experiments, Academic Press, New York

Forrest, S. (1993). Genetic algorithms: Principles of natural selection applied to computation, Science, 261, 872-878 crossref(new window)

Haines, L. M. (1987). The application of the annealing algorithm to the construction of exact optimal designs for linear regression models, Technometrics, 37, 439-447 crossref(new window)

Hamada, M., Martz, H. F., Reese, C. S. and Wilson, A. G. (2001). Finding near-optimal Bayesian experimental designs via genetic algorithms, The American Statistician, 55, 175-181 crossref(new window)

Hartley, H. O. (1959). Smallest composite designs for quadratic response surfaces, Biometric, 15, 611-624 crossref(new window)

Heredia-Langner, A., Carlyle, W. M., Montgomery, D. C., Borror, C. M. and Runger, G. C. (2003). genetic algorithm for the construction of D-optimal designs, Journal of Quality Technology, 35, 28-46

Hoke, A. T. (1974). EconomicaI second-order designs based on irregular fractions of the factorial, Technometrics, 16, 375-384 crossref(new window)

Huang, Y. C. and Wong, W. K. (1998). Sequential construction of multiple-objective optimal designs, Biometrics, 54, 1388-1397 crossref(new window)

JMP Software (2004). Version JMP 5.2. Cary, NC

Karlin, S. and Studden, W. J. (1966). Optimal experimental designs, Annals of Mathematical Statistics, 37, 783-815 crossref(new window)

Kiefer. J. (1959). Optimum experimental designs, Journal of the Royal Statistical Society, Series B, 21, 272-319

Kiefer, J. (1961). Optimum designs in regression problems, Annals of Mathematical Statistics, 32, 298-325 crossref(new window)

Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems, Annals of Mathematical Statistics, 30, 271-294 crossref(new window)

Lauter, E. (1974). Experimental planning in a class of models, Mathematische Operarionsforsh und Statistics, 5, 673-708

Lauter, E. (1976). Optimal multipurpose designs for regression models, Mathematische Operations-forsh und Statistics, 7, 51-68 crossref(new window)

Lee, C. M. S. (1988). Constrained optimal designs, Journal of Statistical Planning and Inference, 18 377-389 crossref(new window)

Lucas. J. M. (1974). Optimum composite designs, Technometrics, 16, 561-567 crossref(new window)

Lucas, J. M. (1976). Which response surface is best, Technometrics, 18, 411-417 crossref(new window)

Meyer, R. K. and Nachtsheim, C. J. (1995). The coordinate-exchange algorithm for constructing exact optimal experimental designs, Technometrics, 37, 60-67 crossref(new window)

Mitchell, T. J. (1974). An algorithm for the construction of D-optimal experimental designs, Techno-metrics, 16, 203-210 crossref(new window)

Mitchell, T. J. and Bayne, C. K. (1978). D-optimal fractions of three-level factorial designs, Techno-metrics, 20, 369-380 crossref(new window)

Montepiedra, G., Myers, D. and Yeh, A. B. (1998). Application of genetic algorithms to the construc-tion of exact D-optimal designs, Journal of Applied Statistics, 25, 817-826 crossref(new window)

Myers, R. H. and Montgomery, D. C. (2002). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York

Myers, R. H., Vining, G. G., Giovannitti-Jensen, A. and Myers, S. L. (1992). Variance dispersion properties of second order response surface designs, Journal of Quality Technology, 24, 1-11

Notz, W. (1982). Minimal point second order designs, Journal of Statistical Planning and Inference, 6, 47-58 crossref(new window)

Park, Y. J., Richardson, D. E., Montgomery, D. C., OzoI-Godfrey, A., Borror, C. M. and Anderson-Cook, C. M. (2005). Prediction variance properties of second-order designs for cuboidal regions, Journal of Quality Technology, 37, 253-266

Stigler, S. M. (1971). Optimal experimental design for polynomial regression, Journal of the Ameri-can Statistical Association, 66, 311-318 crossref(new window)

St. John, R. C. and Draper, N. R. (1975). D-optimality for regression designs: A review, Technometrics, 17, 15-23 crossref(new window)

Welch, W. J. (1982). Branch and bound search for experimental designs based on D-optimality and other criteria, Technomerrics, 1, 41-48 crossref(new window)

Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs, Annals of Mathematical Statistics, 41, 1655-1664 crossref(new window)