Mixture with five components and constraints: an emulsion paint

Step 2 / The region

What the constraints leave of the mixture space

Five shares adding up to one span a four-dimensional body. Without bounds it would be a regular simplex, and the classical simplex designs would fit. The lower and upper bounds cut a body with 24 vertices out of it; the sum and the ratio constraint cut once more and leave 28 vertices. DoEStat computes this body from the constraints as entered — the ratio constraint is written as what it is, a straight line: binder − 1.6 · pigment ≥ 0.

The vertices, the midpoints of edges and faces and the centroid give 124 candidate points. Running all of them would be a waste: a quadratic mixture model for five components has 15 coefficients.

Step 4 / Models

Scheffé models: linear, and with blending terms where needed

Mixture models have no constant — it is contained in the linear terms, because the shares add up to one. The linear terms describe how each component acts on its own; the blending terms (products of two components) describe where two components together do more or less than the sum.

The table is a warning. For all three responses the full quadratic model has the higher adjusted R² — but the lower predicted one, for the wet scrub loss as low as 0.34. Ten blending terms on 25 runs are too many: the model follows the noise. The answer lies in between. DoEStat starts from the quadratic model, works backwards and keeps only the blending terms that are significant; the linear terms always stay.

Two to four blending terms per response are enough. The predicted R² is now above 0.91 for all three, and the lack of fit is nowhere significant.

The coefficients are in pseudo-components: every component is rescaled so that 0 is its lower bound. A linear coefficient is then the predicted response at a corner of the feasible region — a recipe one could actually mix. In real shares it would be the prediction for “100% additive”, a number without meaning. The predictions themselves are the same in both notations.

Analyses of variance of the three models

In the analysis of variance of a mixture the linear terms appear as one line: asking whether a single linear coefficient is zero would be asking whether the pure component has a response of zero. The sensible question is whether the response depends on the recipe at all.

Source	Sum of squares	df	Mean square	F	p
Model	31340	6	5223	933.39	< 0.0001
Linear mixture	30959	4	7740	1383.07	< 0.0001
Pigment × Filler	138.6	1	138.6	24.77	< 0.0001
Water × Additive	275.2	1	275.2	49.18	< 0.0001
Residual	100.7	18	5.596
Lack of fit	67.34	14	4.810	0.58	0.8019
Pure error	33.39	4	8.348
Total	31440	24

Source	Sum of squares	df	Mean square	F	p
Model	36.26	8	4.533	69.88	< 0.0001
Linear mixture	34.66	4	8.666	133.60	< 0.0001
Binder × Pigment	0.8287	1	0.8287	12.78	0.0025
Pigment × Filler	1.513	1	1.513	23.32	0.0002
Pigment × Water	0.9211	1	0.9211	14.20	0.0017
Pigment × Additive	0.5558	1	0.5558	8.57	0.0099
Residual	1.038	16	0.06486
Lack of fit	0.7329	12	0.06107	0.80	0.6573
Pure error	0.3049	4	0.07623
Total	37.30	24

Source	Sum of squares	df	Mean square	F	p
Model	730.6	6	121.8	91.86	< 0.0001
Linear mixture	707.2	4	176.8	133.36	< 0.0001
Pigment × Filler	12.50	1	12.50	9.43	0.0066
Filler × Water	6.280	1	6.280	4.74	0.0431
Residual	23.86	18	1.326
Lack of fit	21.67	14	1.548	2.83	0.1630
Pure error	2.192	4	0.5479
Total	754.5	24

Step 6 / Optimisation

The best feasible recipe

As in the other examples, desirability combines the three goals into one number. The difference: the search is not in a cube but in the body with 28 vertices — every recipe examined must keep all bounds and both linking constraints and add up to 100%.

The overall desirability is 0.83. The viscosity hits its target exactly. The recipe uses the region to the full: additive at its lower bound, pigment just below its upper bound, the solids sum at 38.5% close to the 40% allowed. The ratio of binder to pigment, at 1.90, is well above the minimum — the large amount of binder lowers the wet scrub loss.

Cross-check

What is really in the data

The measurements come from these three models. The letters stand for the deviation of the share from a reference recipe (binder 40%, pigment 20%, filler 15%, water 20%, additive 5%), written as a fraction: b for binder, p for pigment, f for filler, w for water, a for additive.

Viscosity = 100 + 40·b + 220·p + 180·f − 380·w + 1100·a − 3000·w·a + 1200·p·f  (σ = 2.0)
Hiding power = 96.5 − 4·b + 30·p + 6·f − 8·w − 250·p² + 40·p·f  (σ = 0.25)
Wet scrub loss = 14 − 60·b + 25·p + 45·f + 20·w + 300·f² + 150·b·f  (σ = 1.0)

For the viscosity the analysis found exactly the two blending terms that exist: pigment × filler and water × additive. For the hiding power the truth lies in a square of the pigment — the saturation the trace plot shows. A Scheffé model knows no squares; because the shares add up to one, p² appears there as the blending terms of the pigment with all the other components. Exactly those four the analysis kept. For the wet scrub loss the decomposition is less clear-cut: two blending terms with the filler were found, which reproduce the true filler square and its interaction with the binder well within the region studied but are not the same terms.

What counts is the prediction. At the optimal recipe the true values are 100.2 KU, 97.60% and 8.6 µm — predicted were 100.0, 97.37 and 10.1. All three lie within the prediction interval; for the wet scrub loss reality is somewhat better than the prediction.