## Linear programming with a matrix of decision variables [closed]

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.

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en.wikipedia.org/wiki/… – Federico Poloni May 9 2011 at 8:29
Paul, I'm afraid this question is outside the scope of MathOverflow. Federico Poloni has answered your question, but if you want a more elaborate explanation, I suggest you ask at math.stackexchange.com – S. Carnahan May 10 2011 at 21:18