Don't confuse Mixture DOE with Factorial DOE and RSM -

Don’t confuse Mixture DOE with Factorial DOE and RSM

When undertaking formulation work don’t confuse Mixture DOE’s with Factorial/RSM DOE’s

A potential error among DOE practitioners undertaking formulation work is to confuse Mixture DOE with Factorial/RSM DOE’s. This can lead to unsatisfactory or misleading outcomes. Formulation DOE’s are typically undertaken in the food, chemical, petrochemical, paint, cosmetics, and pharmaceutical industries.

The key differences between the two broad types of DOE’s are as follows:

In Factorial/RSM the factor levels are set completely independent of each other. Examples of the factors could be temperature, speed, type of material.
In formulation and mixture DOE’s instead of factors we have ingredients. The equivalent of the levels in Factorial DOE will be the proportions of the ingredients in Mixture DOE. Because the total of the proportions of the ingredients will be the same for each experimental run, if the proportion of one ingredient is changed then the proportions of at least one other ingredient has to be changed to compensate; i.e. unlike the independence of the factor levels in Factorial/RSM, in Mixture the proportions of the ingredients are interdependent.

Consider the two examples that follow. Compare the worksheet layout in Table 2 and Table 3. Note that in Table 2 for the factorial RSM design the factor levels are independent of each other. In the Paint formulation mixture DOE represented by the worksheet of Table 3, it will be observed that for each experimental run the proportions of each ingredient always adds up to 100. If the proportion of one ingredient changes from run to run, the proportion of at least one of the other ingredients must change to compensate. This is a classic example of a mixture DOE. If it was attempted to design the paint hardness formulation as a factorial RSM, with the ingredients treated as factors the conclusions drawn from the experiment would be incorrect.

Example 1 A Factorial/RSM DOE

In this RSM example, the response is conversion % in a chemical process. The factors and levels are set out in Table 1. The worksheet layout is shown in Table 2 and the contour and 3D response surface are illustrated in Graph 1 and Graph 2 respectively.

Table 1

Table 2

Graph 1 Contour plot for RSM Design – Factor No 3, Catalyst is set at 2% for this display

Graph 2 3 D Surface – Factor No 3, Catalyst is set at 2% for this display

Example 2 A Mixture DOE

An experimenter is trying to optimize the formulation of automotive clear coat paint. One of the responses is solids, which the customer wants below 30. The clear coat is a three component mixture, consisting of a monomer, a crosslinker, and a resin. There are constraints on the component proportions:

5 ≤ A (Monomer) ≤ 25
25 ≤ B (Crosslin ker) ≤ 40
50 ≤ C (Re sin) ≤ 70.

A + B + C = 100

The worksheet layout is shown in Table 3. The contour plot and 3D response surface are illustrated in Graph 3 and Graph 4 respectively.

Table 3