Optimizing a linear equation with multiple variables

Question

My maths are a bit rusty, so please excuse me if this is too trivial or has been asked before (very likely!)

I have a set of 20 liquids, each described by three characteristics (A, B and C). How can I mix them to obtain a desired value of A, B and C?

For instance:

• liquid 1 is 2A, 2B, 10C
• liquid 2 is 10A, 5B, 1C
• liquid 3 is 2A, 8B, 9C

...

• liquid 20 is 1A, 3B, 6C

We want a mixture where their linear combination makes 5A, 5B, 5C. We assume that, for instance, mixing liquid 1 with liquid 2 in ratio 1:1 would give us (6A, 3.5B, 5.5C)

I hesitated to post this in a programming forum, because I guess at some point there would be need to rank the different solutions according to how close they are to the target, but it looks to me like solving a set of linear equations (except we might have 0, or many answers...) Could you help me get started?

Answer 1

Never rusty I am going to divide the answer in two( i didn't get completely your question)

You have two liquids

oil=A(1)*t+B(1)*x+C(1)*z;

shampoo=A(2)*t+B(2)x+C(2)*z;

I suppose that you have a set of data for each equation

1) Supposing that you want a function for each liquid with specific values of A, B, C

You can run fminsearch or similar optimisation function(if you have access to Matlab or octave)

what you have to do is to combine the results in a single function you have to create a function that has the values of t,x,z for all your measures and depend on your wanted values A, B, C on this case a vector or a matrix:

function y=liquids([A,B,C],t,x,z)



y1=equation1(A(1),B(1),C(1))=A(1)*t+B(1)x+C(1)*z
y2=equation2(A(2),B(2),C(2))=A(2)*t+B(2)x+C(2)*z

y=y1+y2

if you run an optimisation, you will get the matrix values for each equation

2)

if you want a unique value of A, B, C you have to consider A, B, C on each equation the same.