Mixture Design and D-Optimal Design Online: Formulation Experiments
What Is a Mixture Design?
A mixture design is used for formulation experiments in which the response depends only on the proportions of the ingredients, not on the total amount, and all proportions sum to 1 (sum of xi = 1). Because the proportions are constrained by collinearity, ordinary factorial designs do not apply; instead you use simplex lattice or simplex centroid designs to place the experimental runs, and the feasible space is a simplex rather than a cube. For three or more components, triangle or contour plots show how the response changes with the formulation.
Simplex Lattice Designs
A simplex lattice design places runs on evenly spaced grid points of the simplex: a {3, m} lattice for three components (for m = 2, the endpoints and midpoints) and a {4, m} lattice for four components. The design matrix satisfies the sum-to-one constraint and is fitted with Scheffe polynomials (linear, quadratic or cubic mixture models) without an intercept term. The tool generates lattice plans from the number of components and model order, and supports screening feasible points under upper and lower bounds so that runs fall only inside the feasible region.
D-Optimal Designs
When a mixture has upper and lower bounds (for example, a component must stay between 5% and 30%), the feasible region becomes an irregular simplex and standard lattice designs are no longer appropriate. Optimal designs are then used: D-optimality selects runs from a set of candidate points to maximize the determinant of the information matrix |X'X|, minimizing the variance of the parameter estimates. I-optimality instead minimizes the average prediction variance and suits prediction-oriented objectives. The tool selects runs from the candidate set automatically and lets you specify the candidate-point scale.
How to Use It (Step by Step)
Log in, enter the number of components, their upper and lower bounds and the model order (linear, quadratic or cubic), and choose the design type (lattice or D-optimal). The tool generates an experimental plan in which the proportions sum to 1; run the formulations, enter the responses, fit the mixture model, and view contour or triangle plots with predicted optimal formulas. Mixture experiments still need replication and randomization, and do not skip model significance and lack-of-fit tests or the prediction interval of the optimal formula.