Is it possible to use linear programming when the decision variables are not simply a vector? For example, say I have $n$ countries and $m$ different potential charities I can donate to. The benefits per dollar spent are different for each charity-country pair, and are stored in the $m\times n$ matrix $b$ (if the charity $i$ does not exist in country $j$ then $b_{ij} = 0$). Assume I have a budget of $B$, $x_{ij}$ denotes how much money I donate to charity $i$ in country $j$ and I want to maximize the benefit.
I can formulate this problem as:
$\max_{x} \sum_i \sum_j b_{ij} x_{ij}$
subject to $\sum_i\sum_j x_{ij} \leq B$
[Assume obviously that I will add other constraints along the way (certain amounts donated in each country, maximum donated to each different charity etc), I am just setting this up trivially to begin with to determine if it is doable.]
So rather than just setting up the simplex tableau with the vector $(x_1, x_2, ..., x_l)$ as the bottom row, I actually have an $m \times n$ matrix of decision variables. Is this solvable using linear programming? I am not interested in using a solver, I want to understand the basics of how a problem like this would be solved.
