Keywords: optimization, genetic algorithm, Latin Hypercube, design of experiments.

The choice of location of the evaluation points or plan points is important in getting a good approximation of a system's response, especially when evaluations are expensive. Space-filling designs of experiments (DOE) can be used to specify the points so that as much of the design space is sampled as possible with the minimum number of response evaluations. One popular technique is the optimal Latin hypercube (OLH) design of experiments. However, its generation is non-trivial, time consuming and is - but for the simplest problems - infeasible to carry out by enumeration. Therefore, solving this problem requires an optimization technique to search the design space. As the problem is discrete, it is ideally suited to the use of discrete optimization techniques such as genetic algorithms (GAs).

A method has been developed for formulating the OLH DOE using the Audze-Eglais objective function [1]. It has been shown that the formulation of OLHs is ideally suited to using permutation form of GA [2,3] since the problem uses discrete design variables and the LH requires that values representing DOE levels in a chromosome are not repeated.

One of the shortcomings of LH DoEs is that it is not possible to have points at all the extremities (corner points) of the design space due to the rule of one point per DOE level. This is often desirable in practical design optimization problems. Furthermore, it may be the case that points in the design space are pre-designated, e.g. they have been evaluated in a prior investigation. The developed permutation GA has been extended to account for such situations.

Another shortcoming of LH DOEs is that it is not possible to have different numbers of levels in different design variables. In many practical problems design variables are defined on discrete sets which do not necessarily contain the same number of possible values for each of the individual variables. In such a case, a conventional Latin hypercube DOE is not applicable as some levels in some of the design variables would have more than one point. A strategy has been developed for the treatment of such problems. This addresses a limitation of the standard convention of a Latin hypercube design whilst preserving the uniformity property.

Overall, the permutation GA is shown to be an effective tool for developing OLH DOE and that the extensions described considerably increase the functionality of the tool.

1

P. Audze and V. Eglais, "New approach for planning out of experiments", Problems of Dynamics and Strengths, 35, 104-107, Zinatne Publishing House, Riga, 1977.

2

Z. Michalewicz, "Genetic algorithms + data structures = evolution programs", Springer-Verlag, 1992.

3

S.J. Bates, J. Sienz and V.V. Toropov, "Formulation of the optimal Latin hypercube design of experiments using a permutation genetic algorithm", Proceedings of. 45th AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics & Materials Conf., Palm Springs, California, 19-22 April 2004.

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