Solutions for composition analysis problem

Hi,

We are investigating recycled construction material separated in several size fractions. These consist of a mix of materials such as concrete, ceramics, gypsum, etc. One of the ideas is to make a model to estimate the material content of the size fractions as follows:

a) Composition analysis on the size fractions (data in terms of component metal oxides, e.g. CaO)

b) Composition analysis on the materials

c) Reconstruction of size fraction composition (a) using the composition of materials (b); Size fraction X = a * Concrete + b * Ceramics + c * gypsum + ... , which shall be performed for each metal oxide to obtain the full spectrum

Being quite the MATLAB leek, the question is: Is there an optimisation method available in MATLAB which returns to me the coefficients a, b, c, etc. in such a way that they yield the best approximation of the composition analysis of size fractions using composition analysis of component materials.

Any more required information I will gladly provide!

Thank you very much in advance.

Julian

[rough93]

To make sure I understand the question: You are going to find out how much of each metal oxide is in each material and then each size fraction in order to gain a sum total of the amount of metal oxide of each type across all material? In this case, your a, b, c, d...etc. should all add up to 1 (100%)?

[Julian Stapper]

Yes exactly! The analysis makes the assumption that our list of materials is exhaustive, so the material contents a, b, c, etc. would add up to 1.

[rough93]

I think I understand what you're going for, see my answer below.

Answer 1

[rough93]

Gotcha, so for this application, you'll need to either know the metal oxide content of each material ahead of time through analysis testing or as standard info. and pull it into MATLAB. I'd set it up like this:

el1Concrete = 23.2; %percent of metal oxide 1 in concrete
el2Concrete = 76.8; %percent of metal oxide 2 in concrete
el1Ceramics = 100;  %percent of metal oxide 1 in ceramics
%etc

Then, you'll need your size fractions and how much material is in each (through analysis, as you stated):

Size1Concrete = 5;  %5 units of concrete in size1
Size1Ceramics = 2;  %2 units of ceramics in size1
Size2Concrete = 18; %18 units of concrete in size1

For each size, you can perform calculations in order to find your X equation for each size fraction, then add them together for an overall answer.

TotalSize1 = Size1Concrete + Size1Ceramics; % total amount of units in size 1
X1 = el1Concrete * (Size1Concrete/TotalSize1) + el1Ceramics * (Size1Ceramics/TotalSize1);
TotalSize2 = Size2Concrete + Size2Ceramics; % total amount of units in size 2
X2 = el2Concrete * (Size2Concrete/TotalSize2) + el2Ceramics * (Size2Ceramics/TotalSize2);
X = X1 + X2;

Is this what you're looking for? I don't think you need to use any analytics methods to approximate the answer here, or at least if I understand your request correctly.

[Julian Stapper]

Thank you very much for the answer! The goal is a little more complicated however, but my description may have been a little vague. What we have found, as a simplified example, is that size fraction A contains for 20% of Oxide 1, 30% of Oxide 2, & 50% of Oxide 3, whereas we have also found that it contains the materials Concrete and Ceramics, with known Oxide contents. Now we wish to employ MATLAB to calculate for us the most likely content of these materials (say, 30% Concrete and 70% Ceramics) based on what we found in the size fraction itself and its component materials. Is there any method or algorithm suitable to this situation?

[rough93]

So based on your findings of 20% O1, 30% O2, 50% O3, you want to use MATLAB to tell you "the most likely makeup for this combination of oxides is Concrete + Ceramics"?

If you're looking to solve for roots, there are also numerous root-finding algorithms, however if there are multiple roots or several combinations of oxides that can result in your analytical findings, your root may diverge and your calculations get a little funky.

[Julian Stapper]

Indeed, the most likely % of Concrete and % of Ceramics that are present in this size fraction, which MATLAB should determine using the individual metal oxide spectra of these two materials. Is a root-finding algorithm the appropriate solution for this problem?

I do believe that it will be a tricky method, because there are in the actual situation 4 materials and 5 categories of metal oxides which we are using ("Al", "Si", "Fe", "Ca", and "other" oxides) but we are interested to try it and see where it leads, in any case we have the real results to validate the method.