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Bounds for Maximin Latin Hypercube Designs

Author

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  • Edwin R. van Dam

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands)

  • Gijs Rennen

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands)

  • Bart Husslage

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands)

Abstract

Latin hypercube designs (LHDs) play an important role when approximating computer simulation models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maximin LHDs can be quite time consuming when the number of dimensions and design points increase. In these cases, we can use heuristical maximin LHDs. In this paper, we construct bounds for the separation distance of certain classes of maximin LHDs. These bounds are useful for assessing the quality of heuristical maximin LHDs. Until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs, i.e., for maximin designs without a Latin hypercube structure. The separation distance of maximin LHDs also satisfies these “unrestricted” bounds. By using some of the special properties of LHDs, we are able to find new and tighter bounds for maximin LHDs. Within the different methods used to determine the upper bounds, a variety of combinatorial optimization techniques are employed. Mixed-integer programming, the traveling salesman problem, and the graph-covering problem are among the formulations used to obtain the bounds. Besides these bounds, also a construction method is described for generating LHDs that meet Baer's bound for the (l-script) (infinity) distance measure for certain values of n .

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:57:y:2009:i:3:p:595-608
    DOI: 10.1287/opre.1080.0604
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    1. Husslage, B.G.M. & Rennen, G. & van Dam, E.R. & den Hertog, D., 2008. "Space-Filling Latin Hypercube Designs For Computer Experiments (Revision of CentER DP 2006-18)," Discussion Paper 2008-104, Tilburg University, Center for Economic Research.
    2. Volgenant, A., 1990. "Symmetric traveling salesman problems," European Journal of Operational Research, Elsevier, vol. 49(1), pages 153-154, November.
    3. den Hertog, Dick & Stehouwer, Peter, 2002. "Optimizing color picture tubes by high-cost nonlinear programming," European Journal of Operational Research, Elsevier, vol. 140(2), pages 197-211, July.
    4. 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.
    5. Volgenant, Ton & Jonker, Roy, 1982. "A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation," European Journal of Operational Research, Elsevier, vol. 9(1), pages 83-89, January.
    6. Stinstra, E., 2006. "The meta-model approach for simulation-based design optimization," Other publications TiSEM 713f828a-4716-4a19-af00-e, Tilburg University, School of Economics and Management.
    7. Edwin R. van Dam & Bart Husslage & Dick den Hertog & Hans Melissen, 2007. "Maximin Latin Hypercube Designs in Two Dimensions," Operations Research, INFORMS, vol. 55(1), pages 158-169, February.
    8. Husslage, B.G.M., 2006. "Maximin designs for computer experiments," Other publications TiSEM 216147a0-9c82-48d5-8913-6, Tilburg University, School of Economics and Management.
    9. Driessen, L., 2006. "Simulation-based optimization for product and process design," Other publications TiSEM 51ceee70-3c0a-44a6-a2bd-5, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Rennen, G. & Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2009. "Nested Maximin Latin Hypercube Designs," Discussion Paper 2009-06, Tilburg University, Center for Economic Research.
    2. Rennen, G., 2008. "Subset Selection from Large Datasets for Kriging Modeling," Other publications TiSEM 9dfe6396-1933-45c0-b4e3-5, Tilburg University, School of Economics and Management.
    3. Jing Zhang & Jin Xu & Kai Jia & Yimin Yin & Zhengming Wang, 2019. "Optimal Sliced Latin Hypercube Designs with Slices of Arbitrary Run Sizes," Mathematics, MDPI, vol. 7(9), pages 1-16, September.
    4. Mu, Weiyan & Xiong, Shifeng, 2018. "A class of space-filling designs and their projection properties," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 129-134.
    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. Edwin Dam & Bart Husslage & Dick Hertog, 2010. "One-dimensional nested maximin designs," Journal of Global Optimization, Springer, vol. 46(2), pages 287-306, February.
    7. Rennen, G. & Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2009. "Nested Maximin Latin Hypercube Designs," Other publications TiSEM 1c504ec0-f357-42d2-9c92-9, Tilburg University, School of Economics and Management.
    8. Liuqing Yang & Yongdao Zhou & Min-Qian Liu, 2021. "Maximin distance designs based on densest packings," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(5), pages 615-634, July.
    9. Husslage, B.G.M. & Rennen, G. & van Dam, E.R. & den Hertog, D., 2008. "Space-Filling Latin Hypercube Designs For Computer Experiments (Revision of CentER DP 2006-18)," Discussion Paper 2008-104, Tilburg University, Center for Economic Research.
    10. Rennen, G., 2008. "Subset Selection from Large Datasets for Kriging Modeling," Discussion Paper 2008-26, Tilburg University, Center for Economic Research.
    11. HARCSA Imre Milán & KOVÁCS Sándor & NÁBRÁDI András, 2020. "Economic Analysis Of Subcontract Distilleries By Simulation Modeling Method," Annals of Faculty of Economics, University of Oradea, Faculty of Economics, vol. 1(1), pages 50-63, July.
    12. van Dam, E.R. & Rennen, G. & Husslage, B.G.M., 2007. "Bounds for Maximin Latin Hypercube Designs," Other publications TiSEM da0c15be-f18e-474e-b557-f, Tilburg University, School of Economics and Management.
    13. Husslage, B.G.M. & Rennen, G. & van Dam, E.R. & den Hertog, D., 2008. "Space-Filling Latin Hypercube Designs For Computer Experiments (Revision of CentER DP 2006-18)," Other publications TiSEM 1b5d18c7-b66f-4a9f-838c-b, Tilburg University, School of Economics and Management.
    14. van Dam, E.R., 2008. "Two-dimensional maximin Latin hypercube designs," Other publications TiSEM 61788dd1-b1b5-4c81-9151-8, Tilburg University, School of Economics and Management.
    15. Siem, A.Y.D. & den Hertog, D., 2007. "Kriging Models That Are Robust With Respect to Simulation Errors," Discussion Paper 2007-68, Tilburg University, Center for Economic Research.
    16. 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.
    17. Siem, A.Y.D. & den Hertog, D., 2007. "Kriging Models That Are Robust With Respect to Simulation Errors," Other publications TiSEM fe73dc8b-20d6-4f50-95eb-f, Tilburg University, School of Economics and Management.
    18. Xiangjing Lai & Jin-Kao Hao & Renbin Xiao & Fred Glover, 2023. "Perturbation-Based Thresholding Search for Packing Equal Circles and Spheres," INFORMS Journal on Computing, INFORMS, vol. 35(4), pages 725-746, July.
    19. Tonghui Pang & Yan Wang & Jian-Feng Yang, 2022. "Asymptotically optimal maximin distance Latin hypercube designs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(4), pages 405-418, May.

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