Uniform mixture design via evolutionary multi-objective optimization

Design of experiments is a branch of statistics that has been employed in different areas of knowledge. A particular case of experimental designs is uniform mixture design. A uniform mixture design method aims to spread points (mixtures) uniformly distributed in the experimental region. Each mixture should meet the constraint that the sum of its components must be equal to one. In this paper, we propose a new method to approximate uniform mixture designs via evolutionary multi-objective optimization. For this task, we formulate three M-objective optimization problems whose Pareto optimal fronts correspond to a mixture design of M components (or dimensions). In order to obtain a uniform mixture design, we consider six well-known algorithms used in the area of evolutionary multi-objective optimization to solve M-objective optimization problems. Thus, a set of solutions approximates the entire Pareto front of each M-objective problem, while it implicitly approximates a uniform mixture design. We evaluate our proposed methodology by generating mixture designs in two, three, and up to eight dimensions, and we compare the results obtained concerning those produced by different methods available in the specialized literature. Our results indicate that the proposed strategy is a promising alternative to approximate uniform mixture designs. Unlike most of the existing approaches, it obtains mixture designs for an arbitrary number of points. Moreover, the generated design points are properly distributed in the experimental region.