Advancing QE Skills (20) | Mixture and Formula Optimization: Experimental Design Under Component Constraints

1. Treating Formulas as Ordinary Factors, 27 Trials Yielded Only Waste Data

A manufacturing company sought to improve the formula of a sealant. The quality engineer (QE) followed the approach in a training manual, selecting three levels for each of the three components—resin, curing agent, and filler—and arranged a full factorial experiment with 27 trials. However, the experiment was immediately halted: the fifth trial required 60%, 60%, and 30% of the three components, totaling 150%, which is impossible to formulate. The tester had to "scale down proportionally" to 100% to proceed. When the data was analyzed using regression, the software indicated a singular design matrix. Even after removing the intercept term, the coefficients were contradictory—the resin coefficient was negative, which contradicted the results of previous single-factor experiments, making it impossible to draw a conclusion in the report.

The root cause of such failures lies in a cognitive misunderstanding: the QE assumed that "each factor can independently take levels." However, in a mixture, the sum of all components is always 100%, and the components inherently have a trade-off relationship, where increasing one component reduces the freedom of the others. Mixture design is not just a variant of ordinary DOE; it is an independent method that starts with constraints and even changes the form of the model.

2. Key Principles: Constraints Alter Geometry and Models

The first layer is degrees of freedom. When the sum of q components is always 1, only q-1 components can vary freely. The constant term β0 in ordinary regression overlaps mathematically with the constraint "Σxi = 1." Forcing an intercept term makes the design matrix singular. Therefore, mixture models always use Scheffé canonical polynomials without a constant term:

• Linear model: y = Σβi·xi, number of terms = q;
• Quadratic model: y = Σβi·xi + Σβij·xi·xj, number of terms = q(q+1)/2;
• Special cubic model: adds interaction terms of the form β123·x1x2x3 to the quadratic model, number of terms = q(q+1)/2 + q(q-1)(q-2)/6.

Note that in a quadratic model with three components, there are no xi² squared terms. This is not an omission but a necessity: under the constraint "Σxi = 1," xi² is always equal to xi·(1 - Σ(j≠i)xj), which can be fully absorbed by the linear and pairwise interaction terms. If the software automatically outputs squared terms or someone manually adds them, it indicates that the model form is incorrect.

The second layer is the geometric shape of the feasible region. When components have no upper or lower limits, the feasible region is a complete simplex (triangle/tetrahedron). Once a component is constrained to be "no less than 10% and no more than 60%," the feasible region is cut off by a plane, becoming a polyhedron. The experimental points in mixture design are all arranged at the vertices, midpoints of edges, centroids of faces, and the overall centroid of this polyhedron. The more constraints, the more vertices, and the harder it is to reduce the experimental scale—this is why formula experiments are typically more expensive than process parameter experiments.

The third layer is that optimal solutions often lie within the interior. Mixture responses often exhibit synergistic effects, causing the contour lines to bend into a closed region. Therefore, the output of mixture optimization is not a single "optimal ratio point" but a feasible region that satisfies all constraints—the criteria must focus on "how large the region is and how flat it is" rather than just looking at a peak value.

3. Five Practical Steps and Quantitative Criteria

Step 1: Verify the Feasibility of the Feasible Region and Count the Vertices. List the lower limit Li and upper limit Ui for each component, and check two necessary conditions: ΣLi ≤ 1 and ΣUi ≥ 1. If either condition is not met, the feasible region is empty, and no experimental design will yield a solution. The formula range or components must be adjusted first—this step takes only a few minutes but can save an entire round of experiments. Then estimate the number of vertices to assess the scale: with q components each having upper and lower limits, the number of vertices can reach the order of 2^q, and with 5 components, there could be dozens of vertices. Criteria: if the number of vertices ≤ 25, conduct each trial individually; if it exceeds 25, use D-optimal design to select the most informative subset from the candidate points.

Step 2: Select the Design Type Based on the Number of Components and Constraint Shape.

• 3 components, no constraints: simplex centroid design (7 points) or second-order simplex lattice {3,2} (6 points), with the centroid points repeated 2-3 times, totaling 10-12 trials;
• 4-5 components, no constraints: simplex lattice {q,2}, with 5 components corresponding to 15 points; use {q,3} if special cubic terms need to be estimated;
• Components with upper and lower limits, irregular feasible region: extreme vertices design, supplemented with D-optimal design;
• Components ≥ 6: first group similar raw materials into "major categories" to reduce the number to 4-5, then proceed with mixture design; otherwise, the number of model terms will make the experimental volume uncontrollable.

Two additional scale criteria: the number of experimental points ≥ 1.5 × the number of model terms, and at least 2-3 points should be repeated (otherwise, there will be no pure error, and lack-of-fit tests cannot be performed). Additionally, reserve 3-5 trials for confirmation tests to ensure there is enough margin after the model is completed.

Step 3: Control Execution Deviations, or the Geometry Will Be Useless. Three criteria:

• Weighing accuracy: the actual ratio of each component should differ from the designed ratio by ≤ 0.5% of the target value (relative deviation); otherwise, the ratio deviation will mix with the experimental effects;
• Raw materials: use the same batch number from the same supplier, and treat different batches as blocks, recording batch information in the report;
• Sequence: randomize all trials, and distribute repeated points throughout the experimental sequence; strictly avoid "all repeated points at the end" (environmental drift can be misinterpreted as pure error, leading to inaccurate lack-of-fit tests). Solvents, water, or additives with a fixed total amount in the formula should be treated as pseudo-components—their variations are determined by other components and cannot be adjusted as independent factors or "taken at levels."

Step 4: Validate the Model Using Four-Level Criteria.

• Goodness of fit: adjusted R² ≥ 0.85, and predicted R² (often denoted as Q² or Pred R² in software) ≥ 0.70—focusing only on R² without considering predictive power is the most common self-deception in formula models;
• Lack-of-fit test: p > 0.05; this step can only be performed if there are repeated points;
• Residual diagnostics: normal, no funnel shape, no bending trend. If the lack-of-fit is significant and the residuals show a systematic bend in a certain direction, it indicates that the polynomial degree is insufficient, and the model should be upgraded to a special cubic model (3 components from 6 terms to 10 terms, doubling the number of trials, and should not be the default starting point);
• Prediction accuracy: conduct 3 additional trials near the target region, and the difference between the measured values and the model-predicted values should fall within the prediction interval.

For multi-response optimization, use regional criteria: draw the boundaries for strength, cost, viscosity, etc., and find their intersection. Two criteria:

• The area of the feasible region should be ≥ 5% of the simplex area (a smaller ratio indicates a very narrow window, making mass production likely to fall out of specification);
• The variation in predicted responses within the region should be ≤ target value ± allowable deviation (whether the region is "flat"). A large and flat region is a practical formula.

Step 5: Verification, Standardization, and Re-evaluation Trigger Conditions. Produce independent batches at the center of the optimal region: generally, ≥ 3 batches for regular characteristics, and ≥ 5 batches for key/safety characteristics; the standard deviation between batches should be ≤ 1.5 times the standard deviation during the experimental phase; all critical responses should fall within the specification limits, and the Cpk should be ≥ 1.33. After passing, write the ratio into the BOM and work instructions in the form of "base ratio ± allowable deviation," and specify the mixing method (automatic mixing, weighing verification, poka-yoke). Include the component ratios in the control plan and set the incoming inspection items and monitoring frequency for each component. Four scenarios must trigger re-evaluation:

• Change in raw material supplier or specifications;
• Addition of new raw material sources;
• Change in mixing process;
• Continuous 3 batches of key responses deviating in the same direction.

4. Five Common Misconceptions

Misconception 1: Retaining the Intercept Term or Squared Terms in Mixture Models. Under the constraint "Σxi = 1," the intercept and squared terms cannot be independently estimated. Using ordinary regression templates to fit mixture data often results in "uninterpretable coefficients" or a singular design matrix. Some people simply delete "insignificant" components to force a model, which subtly changes the feasible region.

Misconception 2: Independently Selecting Component Levels and Then Scaling to 100%. In reality, the experiment cannot be conducted as designed. Testers have to make up the difference or add water to adjust, leading to actual ratios that no longer correspond to the designed ratios. Experimental points must be within the feasible region from the start, sacrificing some orthogonality with extreme vertices design rather than "designing one set and executing another."

Misconception 3: Using Standard Simplex Design Despite Constraints. If a component can only be added up to 20% in practice, but the design requires 50%, the experiment will be forced to deviate from the design points. The model is based on "nominal ratios" rather than actual ratios, leading to inaccurate predictions. Criteria: if the deviation rate between design points and process allowable range exceeds 5%, switch to extreme vertices or D-optimal design.

Misconception 4: Conducting Exactly as Many Trials as Model Terms. For a special cubic model with 3 components, there are 10 terms, and someone conducts exactly 10 trials: the residual degrees of freedom are zero, the model perfectly fits each point, R² approaches 1, but it has no predictive power. Criteria: residual degrees of freedom ≥ 3-5, and at least 2 repeated points.

Misconception 5: Directly Mass Producing the "Optimal Point" Without Considering the Region Width and Cost. Focusing only on the peak of the contour lines and not assessing the flatness around it means that even a slight process fluctuation can cause the response to fall out of specification. It also often ignores the cost slope brought by components like resin and precious metals, leading to the selection of a formula with optimal performance but unacceptable cost, which cannot be implemented.

5. Self-Check List

• Verify that Σlower limits ≤ 1 ≤ Σupper limits (non-empty feasible region) and select the design type based on the number of vertices (use extreme vertices if vertices ≤ 25, otherwise use D-optimal).
• All experimental points are within the feasible region, with the actual ratio and designed ratio relative deviation ≤ 0.5%; 2-3 repeated points are distributed throughout the experimental sequence, and raw materials are batched and traced.
• The model uses the Scheffé form without a constant term, with the number of experimental points ≥ 1.5 × the number of model terms, residual degrees of freedom ≥ 3; adjusted R² ≥ 0.85, predicted R² ≥ 0.70, and lack-of-fit p > 0.05.
• After multi-response optimization, the area of the feasible region is ≥ 5% of the simplex area, and the response variation within the region is ≤ target ± allowable deviation; conduct ≥ 3 batches (key characteristics ≥ 5 batches) for confirmation tests and ensure Cpk ≥ 1.33.
• The ratio is written into the BOM and control plan in the form of "base ± allowable deviation," and the re-evaluation trigger conditions for raw material changes, process changes, and continuous deviations are clearly defined.

Constraints define the geometry, and geometry defines the design.

Knowledge code: 6.4.1

Version: v20260930

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping enterprises continuously improve their quality capabilities.