Consider the following linear program $$ \begin{array}{ll} \underset{x_1, x_2} {\text{minimize}} & 50 x_1 + 80 x_2 \\ \text{subject to} & 2 x_1 + 8 x_2 \leq 5 \\ & 6 x_1 + 5 x_2 \leq 10 \\ & x_1, x_2 \geq 0\end{array} $$ Find the optimal cost of this linear program using Fourier-Motzkin elimination.
I have already found the feasible region of this problem to be 0 ≤ x1 ≤ 5/3 and 0 <= x2 <= ⅝:
min 50x1 + 80x2
subj 2x1 + 8x2 ≤ 5
6x1 + 5x2 ≤ 10
x1 ≥ 0
x2 ≥ 0
Rewrite constraints as:
0 ≤ 5 - 2x1 - 8x2
0 ≤ 10 - 6x1 - 5x2
x1 ≥ 0
x2 ≥ 0
=>
0 <= 5 - 2x1
0 ≤ 10 - 6x1
x1 ≥ 0
=>
-5 <= -2x1
-10 <= -6x1
x1 >= 0
=>
5/2 >= x1
5/3 >= x1
x1 >= 0
So the feasible range for x1 is 0 ≤ x1 ≤ 5/3.
To find that the feasible range for x2 is indeed 0 <= x2 <= ⅝, repeat the steps above.
Thus, the constraints aren’t empty, and the problem is feasible.
so I think the optimum solution is (0,0) because this is the minimum of both x1 and x2, but I'm not sure if my logic somewhere is wrong.
