Bounds for Maximin Latin Hypercube Designs

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(infinity) distance measure for certain values of n.