Mixture Designs

Specialized experimental designs for optimizing formulations where ingredients sum to a constant total, focusing on proportional rather than absolute amounts.

Mixture Design Principles & Theory

What they accomplish:

Mixture designs optimize formulations where the factors are proportions or percentages that must sum to unity (100%). The response depends only on the relative proportions, not the absolute amounts, requiring specialized experimental and analysis approaches.

Key Properties:

• • Factors sum to constant (Σxᵢ = 1)
• • Only proportions matter, not totals
• • Simplex-shaped design space
• • Specialized polynomial models
• • Geometric interpretation intuitive

Applications:

• • Chemical formulations
• • Food product development
• • Paint and coating optimization
• • Pharmaceutical blends
• • Concrete mix design

Mixture Design Types & Selection

Standard Mixture Designs:

Simplex-Lattice (q,m):

• Systematic grid of proportions: 0, 1/m, 2/m, ..., 1

• (3,2): Linear model (vertices + edge centers)

• (3,3): Quadratic model (adds interior points)

Simplex-Centroid:

• All possible subsets of q components

• Vertices, edge centroids, face centroids, overall centroid

• Natural for symmetric formulations

Constrained Designs:

• Upper/lower bounds on components: Lᵢ ≤ xᵢ ≤ Uᵢ

• Reduces simplex to irregular polytope

• Requires specialized point selection

Reading the Mixture Design Simplex Plot

Results Interpretation & Optimization

sixsigmacenter.com Implementation Guide

Platform Implementation:

The sixsigmacenter.com platform provides complete mixture design functionality including constraint handling, optimal point selection, canonical mixture model fitting, ternary plots, and multi-response optimization for formulations.

Workflow:

• Define components and any constraints (Lᵢ ≤ xᵢ ≤ Uᵢ)
• Select a lattice, centroid, or constrained design
• Collect response data at design points
• Fit and validate the canonical mixture model
• Generate ternary contour plots and optimize the formulation

Mixture Models & Analysis

Canonical Mixture Models:

Linear (Scheffé):

Linear (Scheffé) mixture model

= expected response for pure component i

• βᵢ = expected response for pure component i

• No intercept term (components sum to 1)

Quadratic:

Quadratic mixture model

βij

= binary blending/interaction coefficient (i < j)

• Includes binary interaction terms

• Captures blending/synergy effects

Special Cubic:

Special cubic mixture model

βijk

= ternary blending coefficient (i < j < k)

• Includes ternary interaction terms

• For complex three-way synergies

Example Data Layout for Simplex-Lattice {3,2} Design

Three-component mixture with proportions summing to 1.0. Vertices test pure components, edges test binary blends, center tests ternary blend. Sample Size Recommendation: {q,m} lattice needs (q+m-1 choose q-1) points plus optional replicates.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.