On uniform design of experiments with restricted mixtures and generation of uniform distribution on some domains

In this paper we propose a new method, based on the conditional distribution method in Monte-Carlo methods, to generate the uniform distribution on the domain T_n(a,b)={(x₁,...,x_n):0≤a _i≤x_i≤b_i≤1,0≤i≤n,x ₁++x_n=1}, where a=(a₁,...,a_n) and b=(b₁,...,b_n). By this new method we can easily obtain uniform designs of experiments with mixtures, i.e., to generate a set of points that are uniformly scattered on the domain T_n(a,b). This approach can apply to generation of uniform distributions on various domains, such as convex polyhedron and simplex. These uniform distributions are useful in experimental design, reliability and optimization.