Determine the equations needed to solve a problem I am trying to come up with the set of equations that will help solve the following problem, but am stuck without a starting point - I can't classify the question to look up more info.
The problem:
Divide a set of products among a set of categories such that a product does not belong to more than one category and the total products within each category satisfies a minimum number.
Example:
I have 6 products that can belong to 3 categories with the required minimums for each category in the final row.  For each row, the allowed categories for that product are marked with an X - eg. Product A can only be categorized in CatX, Product B can only be categorized in CatX or CatY.
$$
        \begin{matrix}
        Product & CatX & CatY & CatZ \\        
        A & X & & \\
        B & X & X & \\
        C & X & & \\
        D & X & X & X \\
        E & & & X\\
        F & & X & \\
        Min Required& 3 & 1 & 2\\
        \end{matrix}
$$
The solution - where * marks how the product was categorized:
$$
        \begin{matrix}
        Product & CatX & CatY & CatZ \\        
        A & * & & \\
        B & * & & \\
        C & * & & \\
        D & & & * \\
        E & & & *\\
        F & & * & \\
        Total & 3 & 1 & 2\\
        \end{matrix}
$$

Let $x_{ij} = 1$ if you put product $i$ in category $j$, $0$ otherwise.  You need
$\sum_i x_{ij} \ge m_j$ for each $j$, where $m_j$ is the minimum for category $j$, 
and $\sum_j x_{ij} = 1$ for each $i$, and each $x_{ij} \in \{0,1\}$.  The last requirement takes it out of the realm of linear algebra.  However, look up "Transportation problem".
