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Generating and improving orthogonal designs by using mixed integer programming

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  • Vieira Jr., Hélcio
  • Sanchez, Susan
  • Kienitz, Karl Heinz
  • Belderrain, Mischel Carmen Neyra

Abstract

Analysts faced with conducting experiments involving quantitative factors have a variety of potential designs in their portfolio. However, in many experimental settings involving discrete-valued factors (particularly if the factors do not all have the same number of levels), none of these designs are suitable. In this paper, we present a mixed integer programming (MIP) method that is suitable for constructing orthogonal designs, or improving existing orthogonal arrays, for experiments involving quantitative factors with limited numbers of levels of interest. Our formulation makes use of a novel linearization of the correlation calculation. The orthogonal designs we construct do not satisfy the definition of an orthogonal array, so we do not advocate their use for qualitative factors. However, they do allow analysts to study, without sacrificing balance or orthogonality, a greater number of quantitative factors than it is possible to do with orthogonal arrays which have the same number of runs.

Suggested Citation

  • Vieira Jr., Hélcio & Sanchez, Susan & Kienitz, Karl Heinz & Belderrain, Mischel Carmen Neyra, 2011. "Generating and improving orthogonal designs by using mixed integer programming," European Journal of Operational Research, Elsevier, vol. 215(3), pages 629-638, December.
    • Handle: RePEc:eee:ejores:v:215:y:2011:i:3:p:629-638
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    References listed on IDEAS

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    1. Kleijnen, Jack P. C., 2005. "An overview of the design and analysis of simulation experiments for sensitivity analysis," European Journal of Operational Research, Elsevier, vol. 164(2), pages 287-300, July.
    2. Appa, G. & Magos, D. & Mourtos, I., 2006. "Searching for Mutually Orthogonal Latin Squares via integer and constraint programming," European Journal of Operational Research, Elsevier, vol. 173(2), pages 519-530, September.
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    4. Edwin Dam & Bart Husslage & Dick Hertog, 2010. "One-dimensional nested maximin designs," Journal of Global Optimization, Springer, vol. 46(2), pages 287-306, February.
    5. Edwin R. van Dam & Gijs Rennen & Bart Husslage, 2009. "Bounds for Maximin Latin Hypercube Designs," Operations Research, INFORMS, vol. 57(3), pages 595-608, June.
    6. Lejeune, Miguel A., 2003. "Heuristic optimization of experimental designs," European Journal of Operational Research, Elsevier, vol. 147(3), pages 484-498, June.
    7. Kuhfeld, Warren F. & Suen, Chung-yi, 2005. "Some new orthogonal arrays," Statistics & Probability Letters, Elsevier, vol. 75(3), pages 169-178, December.
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    Cited by:

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    2. B. McClosky & S. D. Tanksley, 2013. "Optimizing Experimental Design in Genetics," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 520-532, May.
    3. Parker, J.D. & Lucas, T.W. & Carlyle, W.M., 2024. "Sequentially extending space-filling experimental designs by optimally permuting and stacking columns of the design matrix," European Journal of Operational Research, Elsevier, vol. 319(2), pages 600-610.
    4. Guojun Zhang & Quansheng Li & Zhuhe Xu & Yong Zhang, 2022. "Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate," Sustainability, MDPI, vol. 14(16), pages 1-22, August.
    5. Mandal, B.N. & Koukouvinos, C., 2014. "Optimal multi-level supersaturated designs through integer programming," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 183-191.
    6. Besseris, George J., 2012. "Profiling effects in industrial data mining by non-parametric DOE methods: An application on screening checkweighing systems in packaging operations," European Journal of Operational Research, Elsevier, vol. 220(1), pages 147-161.

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