Let's say there are 2 types of TV shows. The first one S1 is usually watched by 2 women and 1 man. The second one S2 is watched by 1 women and 3 men. A company wants to show commercials to reach at least 10 women and 12 men with minimum cost. The cost for a commercial during S1 is 4\$ and during S2 3$.
My LP is as follows:
$x_1:=\#commercialsS1$,$x_2:=\#commercialsS2$, $x_3:=Woman$, $x_4:=Man$
$Minimize\ Z(x_1,x_2) = 4x_1 + 3x_2$
$Subject\ to$
$R_1: x_3 >= 10$, $R_2: x_4 >= 12$
$R_3: 2x_3+x_4 - x_1 = 0$
$R_4: x_3+3x_4 - x_2 = 0$
Somehow it doesn't solve the problem properly.
$R_3$ and $R_4$ seems problematic.
I think we should have $$x_3=2x_1+x_2$$ (Every commersial of $S_1$ would have 2 women and every commercial of $S_2$ would have 1 woman. In fact, you can substitute this in $R_1$.)
Can you express $x_4$ in terms of $x_1$ and $x_2$ using similar reasoning?