Mixture Experiment Design
Mixture Experiment Design (Simplex-Lattice, Simplex-Centroid, D-Optimal) · Also known as: mixture experiment, simplex-lattice design, simplex-centroid design, Scheffé mixture design, Karışım Deneme Deseni (Mixture Design)
Mixture experiment design is a class of constrained experimental design in which the factors are the proportions of components in a blend, subject to the constraint that all proportions sum to one. The framework was formalised by Henry Scheffé in 1958 and covers simplex-lattice, simplex-centroid, and D-optimal mixture designs widely used in pharmaceutical formulation, food science, and materials research.
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When to use it
Use mixture design whenever the experimental factors are component proportions that must sum to a fixed total (typically 1 or 100%). The approach is suitable for formulation optimisation problems in pharmaceutical development (tablet excipients), food science (recipe optimisation), polymer and coating research, and any domain where ingredients are blended. Four assumptions must hold: the mixture constraint Σxᵢ = 1 is exact; a Scheffé polynomial adequately approximates the response surface on the simplex; when lower or upper bounds restrict individual components, a pseudo-component transformation is applied to re-centre the feasible region; and the response is approximately normally distributed and continuous.
Strengths & limitations
- Directly handles the mixture constraint without ad-hoc workarounds, providing geometrically valid design points on the simplex.
- The Scheffé polynomial model separates pure-component effects from blending (interaction) effects, yielding interpretable coefficients.
- Multiple design variants (simplex-lattice, simplex-centroid, D-optimal) let the researcher match design resolution and run count to practical constraints.
- Response-surface contour plots of the simplex give an intuitive visualisation of optimal blend regions.
- The mixture constraint introduces perfect multicollinearity among components, making standard regression inapplicable without reparameterisation.
- When individual components are subject to lower or upper bounds, pseudo-component transformation is required and adds interpretive complexity.
- Scheffé models can require a large number of runs for high-degree or high-component-count designs.
- The method applies only to continuous component proportions; categorical or discrete ingredients require hybrid or categorical mixture extensions.
Frequently asked
How is a mixture design different from a response surface design?
In a standard response surface design (e.g. central composite or Box-Behnken), factors are independent and can be varied freely within their ranges. In a mixture design the factors are proportions that must sum to 1, so they are inherently dependent. This constraint forces the experimental region to be a simplex rather than a hypercube and requires Scheffé polynomials instead of standard polynomial regression.
Which mixture design type should I use — simplex-lattice, simplex-centroid, or D-optimal?
Simplex-lattice designs place points uniformly across the simplex at specified lattice degrees and are well-suited for exploring the full mixture space. Simplex-centroid designs include the centroid and edge midpoints, supporting estimation of all blending terms up to a given degree. D-optimal mixture designs are generated algorithmically and are best when component bounds or process variables restrict the feasible region to an irregular shape that simplex grids cannot cover efficiently.
What is a pseudo-component transformation and when is it needed?
When individual components are constrained to lie within lower or upper bounds (e.g. 0.10 ≤ x₁ ≤ 0.60), the feasible mixture region is no longer the full simplex. A pseudo-component transformation shifts and scales the constrained region back to a unit simplex, allowing standard simplex design points to be generated within the feasible space. Without this transformation, design points may fall outside the allowed composition range.
How do I choose the polynomial degree for the Scheffé model?
Begin with the linear (first-degree) model and test for lack of fit against the quadratic model. If the F-test for lack of fit is significant, fit the quadratic model and again test against the special-cubic model. Continue until the lack-of-fit test is non-significant or the model becomes over-parameterised relative to the number of runs. A significant cross-product term in the quadratic model indicates synergistic or antagonistic blending between those two components.
Sources
- Scheffé, H. (1958). Experiments with Mixtures. Journal of the Royal Statistical Society, Series B, 20(2), 344–360. DOI: 10.1111/j.2517-6161.1958.tb00299.x ↗
- Cornell, J. A. (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (3rd ed.). Wiley. ISBN: 978-0471393374
How to cite this page
ScholarGate. (2026, June 1). Mixture Experiment Design (Simplex-Lattice, Simplex-Centroid, D-Optimal). ScholarGate. https://scholargate.app/en/experimental-design/mixture-design
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Box-Behnken DesignExperimental design↔ compare
- Central Composite DesignExperimental design↔ compare
- Full Factorial DesignExperimental design↔ compare
- Response Surface MethodologyExperimental design↔ compare