Product Development Engineers Ltd

Second DoE tool release – Mixtures with Process Variables

Mixtures with Process Variables infographic showing MPV Design of Experiments, formulation components, process factors, interaction effects, response optimisation, contour plots and experimental design workflow.

Following the release of our Mixtures DoE Tool, Product Development Engineers Ltd has launched its second subscriber tool — an extension that combines mixture design with process variable optimisation. Subscribers can access the tool here. Where the Mixtures tool focuses on ingredient proportions alone, this tool is for situations where the formulation interacts with how it is made or processed. You read about our pure Mixtures tool here. At $10 per month, or only $39 per year, our DoE subscription represents real value, and with more tools to come at these same all-inclusive prices! You can subscribe to our DoE tools here.

What It Adds

Users who are familiar with our Mixtures tool will recognise the same 10-step workflow. The additional capability lies in the definition of one or more process variables — parameters such as temperature, processing time, pressure, or pH — which sit alongside the mixture components and are varied simultaneously across the experimental design.

Process Variable Definition

After defining mixture components, users specify between one and five process variables, each with two or five discrete levels. The tool constructs a combined design that spans both the mixture space and the process variable space, generating a single integrated run sheet.

Mixture process variable design of experiments run sheet showing three mixture components and three process variables in randomised run order
Step 7 of the Mixture + Process Variables DoE tool — a randomised run sheet combining three mixture components (x1, x2, x3) with three process variables (z1, z2, z3), ready for data collection in the field or laboratory.

Extended Model Types

The mixture model types are the same as those in the Mixtures tool — Linear, Quadratic, Special Cubic, and Full Cubic — but users additionally select a model scope that controls how process variables enter the model. The three scope options are: Scheffé only (process variables ignored, useful for comparison), Scheffé plus PV main effects (an additive model where mixture and process variable effects are independent), and Scheffé multiplied by PV (a full interaction model where each mixture term is crossed with each process variable term). The two selectors are chosen independently, giving a wide range of model structures to suit the complexity of the system under study.

Additional Visualisations

Three plot types are added beyond those available in the Mixtures tool. The PV Main Effects plot shows the marginal effect of each process variable on the response across its range, with mixture composition held constant.

PV Main Effects plot showing the marginal effect of three process variables z1, z2 and z3 on response y in a mixture process variable design of experiments
PV Main Effects plot for response y — z1 and z3 both show a positive effect across their range, while z2 has a negative effect. The vertical black line marks the reference (centre) point for each process variable.

The Mix×PV Interaction plot draws separate Cox trace lines for a selected mixture component at each level of a selected process variable, making it easy to see whether the importance of a component changes with processing conditions.

Mixture by PV interaction plot showing Cox trace lines for mixture component x3 at four levels of process variable z2 in a mixture process variable design of experiments
Mixture×PV Interaction plot — Cox trace for component x3 drawn at each level of z2. The consistent separation between the four lines confirms a mixture×process variable interaction: the overall level of response y shifts significantly with z2, while the curved shape of each trace shows that the effect of x3 proportion is nonlinear regardless of processing conditions.

The PV Surface plot shows predicted response as a function of process variable settings at a fixed mixture blend — as a 2D line when one process variable is active, or a 3D surface when two are varied simultaneously.

PV Surface plot showing predicted response y as a 3D surface over two process variables z2 and a second axis in a mixture process variable design of experiments
PV Surface plot for response y — the response rises steeply as z2 moves from +1 toward −1, with the gradient consistent across the second process variable axis. The colour scale from pale orange to deep red reflects the predicted y range of approximately 3 to 6.5.

Optimisation

The optimiser works across both spaces simultaneously, identifying the combination of mixture proportions and process variable settings that maximises, minimises, or hits a target for each response. Multiple responses can be handled using the same desirability framework as the Mixtures tool.

Export

Run sheets and HTML reports include process variable columns alongside mixture component columns, with all additional model terms, interaction plots, and process variable visualisations incorporated into the export.

As with the Mixtures tool, validation testing has been performed against published benchmarks and synthetically generated datasets.

PESL

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