Abstract
Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.
Similar content being viewed by others
References
Angelis L, Bora-senta E, Moyssiadis C (2001) Optimal exact experimental designs with correlated errors through a simulathed annealing algorithms. Comput Stat Data Anal 37:275–296
Anthony DK, Keane AJ (2004) Genetic algorithms for design of experiments on assembled products. Technical report, No. 385, University of Southampton
Aspery SP, Macchietto S (2002) Designing robust optimal dynamic experiments. J Process Control 12(4):545–556
Atkinson AC, Donev AN (1992) Optimum experimental designs. Oxford University Press, Oxford
Babapour Atashgah A (2007) Optimal design of experiments for multi-response linear models. Unpublished Ph.D. thesis, Dep. of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran
Berger MPF, Wong WK (2005) Applied optimal designs. Wiley, New York
Bischoff W (1993) On D-optimal designs for linear models under correlated observations with an application to a linear model with multiple response. J Stat Plan Inference 37:69–80
Borkowski JJ (2003) Using a genetic algorithm to generating small exact response surface designs. J Probab Stat Sci 1(1):65–88
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
Chang SI (1994) Some properties of multi-response D-optimal designs. J Math Anal Appl 184:256–262
Chang SI (1997) An algorithm to generate near D-optimal designs for multiple-response surface models. IIE Trans 29:1073–1081
Chang F, Huang ML, Lin DKJ, Yang H (2001) Optimal designs for dual response polynomial regression models. J Stat Plan Inference 93:309–322
Cook RD, Nachtsheim CH (1980) A comparison of algorithms for constructing exact D-optimal designs. Technometrics 22(3):315–324
Draper NR, Hunter WG (1966) Design of experiments for parameter estimation in multiresponse situations. Biometrika 53:525–533
Drain D, Carlyle WM, Montgomery DC, Borror C, Cook CA (2004) A genetic algorithm hybrid for constructing optimal response surface designs. Qual Reliab Eng Int 20(7):637–650
Fedorov VV (1972) Theory of optimal experiments. Academic Press, New York
Gaffke N, Heiligers B (1995) Algorithms for optimal design with application to multiple polynomial regressions. Metrika 42:173–190
Gaffke N, Mathar R (1992) On a class of algorithms from experimental theory. Optimization 24:91–126
Helmberg C (2002) Semidefinite programming (Invited Review). Eur J Oper Res 137:461–482
Heredia-Langner A, Montgomery DC, Carlyle WM, Borror C (2004) Model-robust optimal designs: a genetic algorithm approach. J Qual Technol 36(3):263–279
Holmstrom K (2004) TOMLAB version 4.4. Tomlab Optimization, San Diego
Imhof L (2000) Optimum designs for a multi-response regression models. J Multivar Anal 72:120–131
Jung JS, Yum BJ (1996) Constructions of exact D-optimal designs by tabu search. Comput Stat Data Anal 21:181–191
Khuri AI, Cornell JA (1996) Response surfaces: designs and analysis, 2nd. edn. Marcel Dekker, New York
Krafft O, Schaefer M (1992) D-Optimal designs for a multivariate regression model. J Multivar Anal 42:130–140
Löfberg J (2005) YALMIP, version 3. Available at http://control.ee.ethz.ch/~joloef/yalmip.php
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Roy SN, Gnanadesikan R, Srivastava JN (1971) Analysis and design of certain quantitative multiresponse experiments. Pergamon Press, New York
Todd MJ (2001) Semidefinite optimization. Acta Numer 10:515–560
Wijesinha MMC (1984) Design of experiments for multiresponse models. Unpublished Ph.D. thesis, Dep. of Statistics, University of Florida, Gainesville
Zellner A (1962) An efficient method of estimating seeming unrelated regressions and tests for aggregation bias. Am Stat Assoc J 57:348–368
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Babapour Atashgah, A., Seifi, A. Optimal design of multi-response experiments using semi-definite programming. Optim Eng 10, 75–90 (2009). https://doi.org/10.1007/s11081-008-9041-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-008-9041-7