John wrote:
> How many of the people reading this have already studied problems of this sort?

Systems Biologists studying metabolic networks use exactly this procedure for constraint-based modeling they call flux balance analysis (FBA).

The first step is to write the stoichiometric coefficients (negative for reactants, positive for products) in a matrix format, where each row represents a reaction and each column represents a compound. For our above example this matrix would thus look like this:

$$ \begin{bmatrix}
p.chip & m.chip & minute & laptop & desktop & profit \\\
-1 & -1 & -4 & 1 & 0 & 0 \\\
-1 & -1 & -3 & 0 & 1 & 0 \\\
0 & 0 & 0 & -1 & 0 & 750 \\\
0 & 0 & 0 & 0 & -1 & 1000

One then computes the constraint space, which corresponds to a polygonal shape in n-dimensional space (where n is the number of compounds) bounded by the linear equations from the matrix. When computed using all reactions in an organism's metabolism, this constraint space corresponds to all metabolic states an organism can be in that don't violate the conservation of mass.

Since this system of equations underdetermined in practice (i.e all points within the boundaries are valid solutions), one needs a further 'objective' function to obtain a unique solution. This objective function is generally a maximization or minimization performed on one or more of the variables. Geometrically, the solution will be located on a vertex, edge, plane, ect. of the constraint space depending on how determined it is. In economics, this usually means maximizing profit (for our example \\( z=750 \text{ laptop} + 1000 \text{ desktop} \\)). In the case of biological organisms, what objective function should be used is less obvious and usually depends on the situation being studied (e.g an organism with sufficient nutrients may try maximizing biomass production, while a starving organism may not).