Practical Application of Mixture Design: From Trial and Error to Optimal Formulation — A Quality Improvement Case Study of a Chemical Company

Abstract: Quality improvement for formulation products (adhesives, coatings, foods, rubbers, etc.) is often fraught with the challenge of "robbing Peter to pay Paul" — increasing one component may improve strength, but it could also lead to increased costs or reduced workability. Mixture Design is specifically designed to address such issues: it incorporates the constraint that the sum of all components must equal 100% into the design, allowing the entire "formulation map" to be drawn with just a few dozen trials, thus identifying the optimal region that balances multiple performance indicators. This article presents a Six Sigma improvement project for an adhesive formulation at a chemical company, detailing the entire process from experimental design, model fitting, to optimal formulation verification.

1. Why Are Formulation Problems So Difficult to Solve?

Quality professionals who have worked on formulation improvements know that it fundamentally differs from optimizing ordinary process parameters. In standard DOE (Design of Experiments), factors such as temperature, pressure, and time are independent; changing one does not affect the others. However, in formulations, the proportions of all components must sum to 100%. If you increase the resin content, the hardener and filler must decrease — the factors are inherently "trade-offs," and none are free to vary independently.

This constraint leads to two consequences. First, trial and error based on experience almost inevitably results in a "robbing Peter to pay Paul" scenario: an engineer at a company increased the resin content from 40% to 50%, which indeed raised the peel strength, but the viscosity doubled, making it impossible to apply, and the customer still returned the product. Reducing the expensive resin to save costs led to the strength falling below the standard line. After half a year of over 30 trials, they still couldn't find a formulation that satisfied all requirements. Second, directly applying standard DOE to formulations can lead to statistical errors: the components are perfectly collinear, and the coefficients in the regression model conflict with each other, making the conclusions unreliable.

Mixture Design is an experimental design method specifically developed to solve such problems. Its core idea is to treat the constraint "the sum of components equals 100%" as a fundamental design condition. All experimental points fall within the simplex (triangle) region, and through specially designed lattice points and centroid points, the mathematical model of the response versus the formulation can be fitted with the fewest number of trials. Mastering this method can transform formulation improvement from a matter of "luck" to a systematic "search."

2. Case Background: The Adhesive Formulation Dilemma of a Chemical Company

In early 2025, a chemical company in East China received continuous complaints from its OEM customers: the peel strength of the epoxy structural adhesive it produced varied significantly between batches, and some batches had a peel strength below the standard value of 8.0 N/mm, causing production line stoppages and rework. The quality department analyzed the shipping data for the past three months and found that the average peel strength was only 8.9 N/mm, with a batch standard deviation as high as 1.6 N/mm — not only was the average close to the standard line, but the variation was also significant.

The main formulation of the adhesive consists of three components: epoxy resin (Component A, determining the primary bonding performance and being the most expensive), hardener (Component B, determining the curing speed and cohesive strength), and toughening filler (Component C, reducing costs and improving toughness, but diluting the bonding strength when added in excess). The sum of these three components is 100%, with a small amount of additives remaining constant. The engineering department had adjusted the formulation over 30 times in the past six months, each time changing only one component: increasing the resin improved the strength, but the viscosity became too high for application; using cheaper fillers saved costs, but the strength dropped below the standard line. More frustratingly, the same formulation performed differently with different batches of raw materials, making it impossible to determine whether the issue was with the formulation or the raw material variability.

The company established this customer complaint as a Six Sigma improvement project, led by a Black Belt. In the Define phase, the problem was identified as "low average peel strength and high variation." In the Measure phase, a measurement system analysis (GR&R) for the peel test was conducted to ensure the testing system was reliable, and data was collected to establish a baseline. In the Analyze phase, a fishbone diagram and regression analysis were used to identify the root causes, confirming that the primary issue was the不合理的比例 and the interaction effects among the three components — each component made sense individually, but they constrained each other when combined. In the Improve phase, the project team decided to introduce Mixture Design to identify the optimal region for the three components in a single step.

3. Three Basic Approaches of Mixture Design

Before diving into the project, it's important to understand the "arsenal" of Mixture Design. Depending on the number of components and the constraints, three common designs are used:

1. Simplex Lattice Design: Denoted as {number of components, order}. For a three-component, second-order lattice design {3,2}, the experimental points include 3 pure component points and 3 binary equal proportion points, totaling 6 trials, sufficient to fit a second-order model. For a third-order lattice design {3,3}, 3 additional "1/3 and 2/3" combination points are added, totaling 10 trials, which can fit a special third-order model.
2. Simplex Centroid Design: The experimental points include all pure component points, all binary centroid points (1/2, 1/2), and the ternary centroid point (1/3, 1/3, 1/3). For a three-component simplex centroid design, there are 7 design points. Including the repeated centroid point, the design typically requires around 10 trials, making it the most commonly used choice for three-component scenarios.
3. Extreme Vertices Design: Used when components have upper and lower limits. For example, if a component cannot be less than 10% or more than 60%, the feasible region is no longer a complete triangle but a "cut" polygon, and the experimental points are taken at the vertices and edges of the polygon.

This project had only three components with no strict upper or lower limits, so the project team chose the simplex centroid design: 7 design points, with the centroid point repeated 3 times, totaling 10 trials. The raw material batches were unified, the mixing process was fixed, and the trials were conducted in a random order to avoid environmental drift affecting the results.

4. Case Progress: 10 Trials to Draw the Formulation Map

The trials were conducted based on the proportions of the three components (X1 resin, X2 hardener, X3 filler), with the response variable being peel strength (N/mm). The results of the 10 trials are as follows:

First, let's look at the information revealed by the data: pure resin (Trial 1) had a strength of 9.8, while pure hardener and pure filler were very low. The binary combinations were significantly higher than their respective pure components, indicating a positive interaction effect. The ternary centroid points (Trials 7-10) had an average strength of 12.3, significantly higher than any binary combination — this suggests that the three components together have a "1+1+1>3" synergistic effect, and the optimal formulation must be within the triangle, not on the edges.

The project team used statistical software to fit a mixture model to the data. Initially, a second-order model was fitted, but the actual measured values at the centroid (12.3) were much higher than the predicted values (around 8.9), indicating significant residuals and a lack of fit. This suggested that the interaction effects among the components exceeded the range that a second-order model could describe. Therefore, a special third-order model (adding X1X2X3 interaction terms) was used. The final model had an R² of 0.98, a non-significant lack of fit test, and random residuals, confirming the model's reliability.

5. Case Analysis: Locking the Optimal Region on a Ternary Plot

After building the model, the project team plotted the predicted results on a ternary plot: the three vertices of the triangle represent pure resin, pure hardener, and pure filler, respectively. Any point inside the triangle represents a formulation, and the contour lines represent the predicted peel strength. This "formulation map" clearly showed the distribution of strengths — the optimal region appeared as an ellipse, roughly falling within the range of X1 resin 0.45-0.55, X2 hardener 0.30-0.40, and X3 filler 0.10-0.20, with a predicted peel strength of 12.0-12.6 N/mm.

However, strength is not the only indicator. The team added two additional constraints to the same plot:

1. Cost Constraint — resin is the most expensive component, and increasing X1 by 10 percentage points raises the raw material cost by about 120 yuan per ton. X1 must be controlled to 0.55 or less.
2. Application Viscosity Constraint — when the filler content is too low, the adhesive becomes too thin and prone to sagging. X3 must not be less than 0.10.

The intersection of these three boundaries defined the feasible region that balanced strength, cost, and workability.

The team ultimately selected the formulation: 50% resin, 32% hardener, and 18% filler, with a model-predicted peel strength of 12.3 N/mm. The cost of raw materials per ton was reduced by about 6% compared to the original formulation. Subsequent independent verification trials measured 12.1, 12.5, and 12.2 N/mm, with an average of 12.27 N/mm, closely matching the prediction, thus confirming the model's reliability.

6. Verification and Standardization: From Optimal Formulation to Batch Stability

After the verification was successful, the project team did not rush to switch to mass production but took three steps to solidify the results:

1. Small Batch Pilot Production Confirmation: Continuous production of 5 batches resulted in an average peel strength of 12.2 N/mm, with the batch standard deviation decreasing from 1.6 N/mm before the improvement to 0.4 N/mm. All batches exceeded the standard value of 8.0 N/mm, providing both the customer and the team with confidence.
2. Update Controlled Documents: The new formulation was written into the formulation sheet and work instruction. The weighing error ranges for the components (resin ±0.5%, hardener ±0.3%, filler ±0.5%) were clearly specified in the control plan, and the weighing data for key components were included in SPC monitoring to prevent individual batches from reverting to the old formulation due to human error.
3. Revise FMEA: "Incorrect formulation ratio" was identified as a high-priority failure mode, and double-checking and electronic scale poka-yoke devices were added. Three months of tracking data showed zero customer complaints and zero returns, saving the company about 1.2 million yuan in quality costs annually. The project's financial benefits were reviewed and approved by the Six Sigma team.

The benefits of this project extended beyond a single production line. During the project review, the team realized that the same method could be applied to other formulation products within the company — the pigment formulation in the coatings workshop, the surfactant blending in the cleaning agent workshop, and even the development of new product prototypes. Within half a year, the team used the same Mixture Design process to optimize two more formulations, with each project from initiation to verification taking only about three weeks, compared to the previous method of trial and error, which took at least two to three months. This is the most valuable aspect of Six Sigma improvement: the project not only yields a specific optimal formulation but also establishes a replicable method for finding the best formulation.

7. Common Pitfalls and Key Points for Implementation

Reflecting on this case, there are five common pitfalls that future practitioners should be wary of:

• Pitfall 1: Directly Applying Standard DOE to Formulations: The sum of components equaling 100% causes collinearity, leading to conflicting regression coefficients and unreliable conclusions. Formulation problems must be addressed using the specialized design points and models of Mixture Design.
• Pitfall 2: Optimizing a Single Response: If only strength is considered, increasing the resin content to over 60% might be the "best" option, but it would lead to increased costs and reduced workability. Use overlapping contour lines or a satisfaction function to balance multiple responses.
• Pitfall 3: Ignoring the Process Limits of Components: When there are upper and lower limits, the feasible region is not a complete triangle. Use Extreme Vertices Design to avoid the "optimal formulation" falling into an unproducible region.
• Pitfall 4: Switching to Mass Production Without Verification: No matter how beautiful the model, it must be verified with independent trials. In this case, the verification trials had an error of less than 0.1 N/mm compared to the prediction, giving the team the confidence to switch to mass production.
• Pitfall 5: Confusing Mixture Variables with Process Variables: When process variables such as stirring speed and curing temperature also affect the response, use a "Mixture-Process Variable" combined design to separately understand the effects of each type of variable.

For implementation, it is recommended to start with small formulations of three components, using about 10 trials to run through the entire "design-modeling-verification" process. Before the trials, ensure that the measurement system is stable to avoid mistaking measurement errors for formulation effects. During the analysis phase, use more ternary contour plots to make the conclusions understandable to process, production, and procurement teams, ensuring that the improvement plan is actionable.

8. Conclusion

The value of Mixture Design lies in its respect for the inherent nature of formulation problems — the components are interdependent and yet synergistic. With just a few dozen small samples and a dozen trials, it can transform "formulation by trial" into "formulation by calculation": first, draw the map, then find the intersection, and finally, verify and standardize. For industries driven by formulations such as adhesives, coatings, foods, rubbers, and cosmetics, Mixture Design is one of the most cost-effective tools in the Six Sigma improvement toolkit — it requires no expensive equipment, only a scientific experimental design and a clear ternary plot. The next time a formulation issue arises, don't rush to adjust the components; start with a Mixture Design.

The essence of Mixture Design is to transform "formulation by trial" into "formulation by calculation," ensuring that every raw material is used to its fullest potential.

Knowledge code: 6.4.1

Version: v20260805

Author: Quality Think Tank Quality Think Tank is dedicated to providing systematic professional knowledge, methodologies, and practical tools to quality management practitioners, helping companies continuously improve their quality capabilities.