Q.
Given the LPP :
Minimize $ f = 2x_1 - x_2 $
$ x_{1}\ge0, x_{2}\ge0 $
$ x_{1}+x_{2}\ge5 $
$ -x_{1}+x_{2}\le1 $
$ 5x_1 + 4x_2 \le40 $
The solution is
AMUAMU 2016Linear Programming
Solution:
Given, minimize $f=2 x_{1}-x_{2}$
$x_{1} \geq 0, x_{2} \geq 0$
$x_{1}+x_{2} \geq 5$
$-x_{1}+x_{2} \leq 1$
and $x_{1}+x_{2} \leq 1$
Graph of $x_1 + x_2 = 5$
$x_1$
0
5
$x_2$
5
0
Putting $(0,0)$ in the inequality $x_{1}+x_{2} \geq 5$
$\Rightarrow 0+0 \geq 5$
$\Rightarrow 0 \geq 5$
(which is false)
$\Rightarrow$ Half plane away from origin.
Graph of $-x_{1}+x_{2}=1$
$\Rightarrow x_{1}-x_{2}=1$
$x_1$
0
-1
$x_2$
1
0
Putting $(0,0)$ in the inequality $-x_{1}+x_{2} \leq 1$
$\Rightarrow 0+0 \leq 1 \Rightarrow 0 \leq 1$
(which is true)
Half plane towards the origin. and graph of $5 x_{1}+4 x_{2}=40$
$x_1$
0
8
$x_2$
10
0
Putting $(0,0)$ in the inequality $5 x_{1}+4 x_{2} \leq 40$
$\Rightarrow 0+0 \leq 40 $
$ \Rightarrow 0 \leq 40$ (which is true)
$\Rightarrow$ Half plane towards the origin.
Here, we obtain feasible region $A B P Q$ s Coordinates of $Q$.
Solving $ x_{1}-x_{2}=-1$
$5 x_{1}+4 x_{2}=40$
We get $x_{1}=4, x_{2}=5, Q \equiv(4,5)$,
Coordinates of $P$.
Solving $ x_{1}-x_{2}=-1 $
and $ x_{1}+x_{2}=5 $
We get $ x_{1}=2, x_{2}=3$
$P \equiv(2,3)$
Corner point
$Z = 2x_1 - x_2$
A (5, 0)
10
B(8, 0)
16
P(2, 3)
1 (minimum)
Q(4, 5)
3
| $x_1$ | 0 | 5 |
| $x_2$ | 5 | 0 |
| $x_1$ | 0 | -1 |
| $x_2$ | 1 | 0 |
| $x_1$ | 0 | 8 |
| $x_2$ | 10 | 0 |
| Corner point | $Z = 2x_1 - x_2$ |
|---|---|
| A (5, 0) | 10 |
| B(8, 0) | 16 |
| P(2, 3) | 1 (minimum) |
| Q(4, 5) | 3 |