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=(x ₁, ., x _q) ^', 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, x ₁, is different from that of the others, as well as its D-optimal design for the entire experimental region.