Hi @Byron_JMP,
I also thought about using Mixture design, but since the total quantity of X1+X2+X3+X4 is not fixed (it should be between 9 and 18 units), I switched to an Optimal factorial design.
The Mixture design can be a good alternative for this use case by adding a 5th continuous factor for the total quantity of component that can vary between 9 and 18 :
![Victor_G_0-1724137302055.png Victor_G_0-1724137302055.png](/t5/image/serverpage/image-id/67283i116957B9AF712851/image-size/medium?v=v2&px=400)
Specifying a Scheffé Cubic model and adding a quadratic effect for X5 (Total Quantity) enables to have an interesting design respecting all the constraints :
![Victor_G_1-1724137372671.png Victor_G_1-1724137372671.png](/t5/image/serverpage/image-id/67284i980E78729D36C7A2/image-size/medium?v=v2&px=400)
(Value of X4 for row25 could be switched to 0 without big changes, to avoid using 4 components in the mixture if really needed).
Script to generate the design :
DOE(
Custom Design,
{Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0, 0.89, "X1", 0 ),
Add Factor( Mixture, 0, 0.89, "X2", 0 ), Add Factor( Mixture, 0, 0.89, "X3", 0 ),
Add Factor( Mixture, 0, 0.22, "X4", 0 ),
Add Factor( Continuous, 9, 18, "Total Quantity", 0 ),
Set Random Seed( 1320459420 ), Number of Starts( 743 ), Add Term( {1, 1} ),
Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),
Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {1, 1}, {4, 1} ), Add Term( {2, 1}, {3, 1} ),
Add Term( {2, 1}, {4, 1} ), Add Term( {3, 1}, {4, 1} ),
Add Term( {1, 1}, {2, 1}, {3, 1} ), Add Term( {1, 1}, {2, 1}, {4, 1} ),
Add Term( {1, 1}, {3, 1}, {4, 1} ), Add Term( {2, 1}, {3, 1}, {4, 1} ),
Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {2, 1}, {3, 1} ), Add Term( {1, 1}, {4, 1} ),
Add Term( {2, 1}, {4, 1} ), Add Term( {3, 1}, {4, 1} ), Add Term( {5, 2} ),
Add Alias Term( {1, 1}, {5, 1} ), Add Alias Term( {2, 1}, {5, 1} ),
Add Alias Term( {3, 1}, {5, 1} ), Add Alias Term( {4, 1}, {5, 1} ),
Set Sample Size( 30 ), Simulate Responses( 0 ), Save X Matrix( 0 ), Make Design}
)
And datatable is attached.
Hope this discussion may help @stat_ranger,
Victor GUILLER
Scientific Expertise Engineer
L'Oréal - Data & Analytics