Question
Maximize Z = 50x + 30y
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$
✓
Answer
First, we will convert the given inequations into equations, we obtain the following equations:
2x + y = 18, 3x + 2y = 34
Region represented by 2x + y ≥ 18:
The line 2x + y = 18 meets the coordinate axes at A(9, 0) and B(0, 18) respectively. By joining these points we obtain the line 2x + y = 18.
Clearly (0,0) does not satisfies the inequation 2x + y ≥ 18.
So,the region in xy plane which does not contain the origin represents the solution set of the inequation 2x + y ≥ 18.
Region represented by 3x + 2y ≤ 34:
The line 3x + 2y = 34 meets the coordinate axes at $\text{C}\Big(\frac{34}{3},0\Big)$ and D(0, 17) respectively. By joining these points we abtain the line 3x + 2y = 34.
Clearly (0. 0) satisfies the inequation 3x + 2y ≤ 34. So, the region containing the origin represents the silution set of the inequation 3x + 2y ≤ 34.
The corner of the feasible region are A(9, 0), $\text{C}\Big(\frac{34}{3},0\Big)$ and E(2, 14).
The values of Z at these corner points are as follows.
Therefore, the maximum value of Z $\frac{1700}{3}$ is at the point $\Big(\frac{34}{3},0\Big)$.
Hence, $\text{x}=\frac{34}{3}$ and $\text{y}=0$ is the optimal solution of the given LPP.
Thus, the optimal value of Z is $\frac{1700}{3}$.
2x + y = 18, 3x + 2y = 34
Region represented by 2x + y ≥ 18:
The line 2x + y = 18 meets the coordinate axes at A(9, 0) and B(0, 18) respectively. By joining these points we obtain the line 2x + y = 18.
Clearly (0,0) does not satisfies the inequation 2x + y ≥ 18.
So,the region in xy plane which does not contain the origin represents the solution set of the inequation 2x + y ≥ 18.
Region represented by 3x + 2y ≤ 34:
The line 3x + 2y = 34 meets the coordinate axes at $\text{C}\Big(\frac{34}{3},0\Big)$ and D(0, 17) respectively. By joining these points we abtain the line 3x + 2y = 34.
Clearly (0. 0) satisfies the inequation 3x + 2y ≤ 34. So, the region containing the origin represents the silution set of the inequation 3x + 2y ≤ 34.
The corner of the feasible region are A(9, 0), $\text{C}\Big(\frac{34}{3},0\Big)$ and E(2, 14).
The values of Z at these corner points are as follows.
|
$\text{Corner point}$
|
$\text{Z}=50\text{x}+30\text{y}$
|
|
$\text{A}(9, 0)$
|
$50\times9+3\times0=450$
|
|
$\text{C}\Big(\frac{34}{3},0\Big)$
|
$50\times\frac{34}{3}+30\times0=\frac{1700}{3}$
|
|
$\text{E}(2, 14)$
|
$50\times2+30\times14=520$
|
Hence, $\text{x}=\frac{34}{3}$ and $\text{y}=0$ is the optimal solution of the given LPP.
Thus, the optimal value of Z is $\frac{1700}{3}$.
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