Q. The objective function $z = x_{1} + x_{2}$, subject to $x_{1} + x_{2} ≤ 10, - 2x_{1} + 3x_{2} ≤ 15, x_{1} ≤ 6, x_{1} , x_{2} ≥ 0$ has maximum value ______________ of the feasible region.
Solution:
Given, objective function, $Z=x_{1}+x_{2}$ and constraints are
$x_{1}+x_{2} \leq 10,-2 x_{1}+3 x_{2} \leq 15, x_{1} \leq 6, x_{1}, x_{2} \geq 0$
The point of intersection of lines $x_{1}+x_{2}=10$
and $-2 x_{1}+3 x_{2}=15$ is $B(3,7)$ and point of intersection of lines $x_{1}=6$ and $x_{1}+x_{2}=10$ is $C(6,4)$
The feasible region is $OABCD$. The corner points of the feasible region are
$O(0,0), A(0,6), B(3,7), C(6,4)$ and $D(6,0)$
At O(0, 0)
Z = 0 + 0 = 0
A(0, 6)
Z = 0 + 6 = 6
B(3, 7)
Z = 3 + 7 = 10
C (6, 4)
Z = 6 + 4 = 10
D (6, 0)
Z = 6 + 0 = 6
Hence , $Z$ is maximum at every point of the segment combined two points $B(3, 7 )$ and $C(6, 4)$
| At O(0, 0) | Z = 0 + 0 = 0 |
| A(0, 6) | Z = 0 + 6 = 6 |
| B(3, 7) | Z = 3 + 7 = 10 |
| C (6, 4) | Z = 6 + 4 = 10 |
| D (6, 0) | Z = 6 + 0 = 6 |