Sachchidanand Prasad

25-05-2024

Problem: Consider the following Linear Programming Problem P \[ \begin{array}{ll} \text{minimize} & \quad 5x_1 + 2x_2 \\ \text{subject to} & \quad 2x_1 + x_2 \leq 2 \\ & \quad x_1 + x_2 \leq 1\\ & \quad x_1, x_2 \geq 0. \end{array} \] The optimal value of the problem P is equal to
  • $5$
  • $4$
  • $0$
  • $2$
linear-programming GATE-2024
Solution: Let $f(x_1,x_2)= 5x_1 + 2x_2$. We will use graphical method to solve this linear programming problem (LPP). Look at the figure below.
LPP problem graph
Now the minimum value of the objective function will be on the axes in the feasible region. Let us substitute the value and see which one is corresponding to the minimum. \begin{align*} f(1,0) & = 5 \\ f(0,1) & = 2 \\ f(0,2) & = 4. \end{align*} Thus, the minimum value of the LPP is $2$ and the corresponding value of point will be $(0,1)$.