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Maximin Latin Hypercube Designs in Two Dimensions

Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
edwin.vandam{at}uvt.nl
b.g.m.husslage{at}uvt.nl
d.denhertog{at}uvt.nl
melissen{at}isa.ewi.tudelft.nl

The problem of finding a maximin Latin hypercube design in two dimensions can be described as positioning n nonattacking rooks on an n x n chessboard such that the minimal distance between pairs of rooks is maximized. Maximin Latin hypercube designs are important for the approximation and optimization of black-box functions. In this paper, general formulas are derived for maximin Latin hypercube designs for general n, when the distance measure is l or l¹. Furthermore, for the distance measure l², we obtain maximin Latin hypercube designs for n 70 and approximate maximin Latin hypercube designs for other values of n. All these maximin Latin hypercube designs can be downloaded from the website http://www.spacefillingdesigns.nl. We show that the reduction in the maximin distance caused by imposing the Latin hypercube design structure is small. This justifies the use of maximin Latin hypercube designs instead of unrestricted designs.

Subject classifications: branch-and-bound; circle packing; Latin hypercube design; mixed-integer programming; noncollapsing; space-filling.
History: Received February 2005; revision received October 2005; accepted October 2005.