06-07-2024
Problem: Solve the following optimization problem using graphical method.
\[
\text{ Maximize } z = 2x_1 + 4x_2
\]
subject to
\begin{align*}
x_1 + 2x_2 & \leq 5 \\
x_1 + x_2 & \leq 4 \\
x_1, x_2 & \geq 0.
\end{align*}
Solution: The graph of the given optimization problem is given below and the maximum is $z = 10$.
Note that, the feasible region is bounded by the polygon $OABC$, and we have
\begin{align*}
z(A) & = Z(4,0) = 8 \\
z(B) & = z(3,1)= 10 \\
z(C) & = z \left(0, \frac{5}{2}\right) = 10.
\end{align*}
Thus, the optimum value of the LPP is $10$, also there are infinitely many solutions as the objective function is parallel to one of the constrain. All th points on the line $x_1 + 2x_2 = 5$ will be a solution to the optimal value $10$.