D-optimal designs for quadratic mixture canonical polynomials with spline

Experiments with mixtures are special types of experiments in which the response depends only on the proportions of the input variables x=(x1,…,xq)′, not on the total amount of ingredients. When a response variable cannot be adequately represented by a single polynomial function of the input variable over the entire experimental region, a possible solution is to use a regression model, termed the mixture polynomial with spline, which consists of grafted polynomial submodels. In this paper, we investigate a special mixture polynomial with spline in which the contribution of one ingredient proportion to the response variable, for example, x1, is different from that of the others, as well as its D-optimal design for the entire experimental region.