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 Tn(a,b)={(x1,...,xn): 0[less-than-or-equals, slant]ai[less-than-or-equals, slant]xi[less-than-or-equals, slant]bi[less-than-or-equals, slant]1,0[less-than-or-equals, slant]i[less-than-or-equals, slant]n, x1+...+xn=1}, where a=(a1,...,an) and b=(b1,...,bn). 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 Tn(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.