It seems from an easy google of "learning linear programming" that a common way of learning it is to work with Matrices that represent "games" for two players. Here is one I have stumbled across.
We have a matrix, $ A = (a_{i,j}) $ where $1 \leq i \leq m $ and $1 \leq j \leq n $
Two players, the row player and the column player play a game with the row player selecting a row and the column player selecting a column.
If $a_{i,j} > 0$ the row player receives a payout amount of $a_{i,j}$. If $a_{i,j}$ < 0 the row player pays a payout amount of $|a_{i,j}|$ to the column player.
The payout matrix A is known to both players.
The Row player picks the i-th row with probability $p_{i}$. Therefore, $p_{1}$ + $p_{2}$ + .. + $p_{m}$ = 1. The problem is to find an optimal strategy for the Row player.
Consider a strategy for the row player where it chooses row i with probability $p_{i}$. If the Column player chooses column j then the average payo for the Row player is:
$p_{1}a_{1,j}$ + $p_{2}a_{2,j}$ + ... + $p_{m}a_{m,j}$
Therefore, the Row-player is guaranteed a payout of at least:
$min_{j}$($p_{1}a_{1,j}$ + $p_{2}a_{2,j}$ + + $p_{m}a_{m,j}$)
So, the Row player has to choose
($p_{1}$, $p_{2}$, ... , $p_{m}$) that maximizes
z = $min_{j}$($p_{1}a_{1,j}$ + $p_{2}a_{2,j}$ + ... + $p_{m}a_{m,j}$):
with the constraints:
$$ \sum_{i=1}^m p_{i} = 1 $$ $$ p_{i} \geq 0 \\\\i \in (1, ...,m) $$
Express this as a linear program. Why is your linear program equivalent to the constraints above?
Anyone have any insight? Or a good place to start?