Question

For the following LP problem, graph the region of feasible solution and solve by the corner-point method.

Maximize z = 5x1 + 8x2
Subject to x1 + x2 ≥ 6
3x1 + 2x2 ≤ 30
2x1 + x2 ≤ 5
x1 , x2 ≥ 0

Answer 1

$x_1+x_2\geq 6$

$3x_1+2x_2\leq 30$

$2x_1+x_2\leq 5$

$x_1, x_2\geq 0$

Maximize $z=5x_1+8x_2$

Green region shows all solutions of $\begin{cases} x_1+x_2\geq 6 \\3x_1+2x_2\leq 30 \\ 2x_1+x_2\leq 5 \end{cases}$

And red region shows all solutions of $x_1,x_2\geq 0$

We can see that there is no solution of the system $\begin{cases} x_1+x_2\geq 6 \\3x_1+2x_2\leq 30 \\ 2x_1+x_2\leq 5 \\ x_1,x_2\geq 0 \end{cases}$ because intersection of these regions is empty.

Therefore, we can’t maximize $z=5x_1+8x_2$ because there is no $(x_1,x_2)$ , for which all inequalities are satisfied.