Solve the linear programming problem by the method of corners.
Find the minimum and maximum of
P = 3x + 2y subject to
3x + 5y ≥ 20
3x + y ≤ 16
−2x + y ≤ 2
x ≥ 0, y ≥ 0
The minimum is P =
at (x, y) =
To solve the linear programming problem by the method of corners, we need to find the coordinates of the corners of the feasible region defined by the given inequalities. Then we can substitute these coordinates into the objective function P = 3x + 2y to determine the minimum and maximum values.
Let's analyze each inequality to determine the region it defines:
1) 3x + 5y ≥ 20:
To find the boundary line, we set the inequality as an equation and solve for y:
5y = 20 - 3x
y = (20 - 3x) / 5
To plot this line, we need two points. Substitute x = 0, we get y = 4.
Substitute x = 4, we get y = 16/5 or 3.2.
2) 3x + y ≤ 16:
To find the boundary line, we set the inequality as an equation and solve for y:
y = 16 - 3x
To plot this line, we need two points. Substitute x = 0, we get y = 16.
Substitute x = 5.3333 (approximately), we get y = 0.
3) -2x + y ≤ 2:
To find the boundary line, we set the inequality as an equation and solve for y:
y = 2 + 2x
To plot this line, we need two points. Substitute x = 0, we get y = 2.
Substitute x = -2, we get y = -2.
Now, let's plot these lines on a graph and shade the region that satisfies all given inequalities:
y
^
|
4 | •
| (0,4)
3.2 |
|
|
2 | •
| (5.33,0)
1.8 |
|
|
0 |___________________________
0 4 8 12 16 --> x
The feasible region is the shaded region on the graph above.
Next, we need to find the corners of the feasible region. This can be done by finding the intersection points of the boundary lines.
1) Intersection of the first and second lines:
(20 - 3x) / 5 = 16 - 3x
20 - 3x = 80 - 15x
12x = 60
x = 5
Substitute x = 5 into the second line equation:
y = 16 - 3(5)
y = 16 - 15
y = 1
Corner point (x, y) = (5, 1)
2) Intersection of the first and third lines:
(20 - 3x) / 5 = 2 + 2x
20 - 3x = 10 + 10x
13x = 10
x = 10/13
Substitute x = 10/13 into the third line equation:
y = 2 + 2(10/13)
y = 2 + 20/13
y = (26 + 20)/13
y = 46/13
Corner point (x, y) = (10/13, 46/13)
3) Intersection of the second and third lines:
16 - 3x = 2 + 2x
5x = 14
x = 14/5
Substitute x = 14/5 into the second line equation:
y = 16 - 3(14/5)
y = 16 - 42/5
y = (80 - 42)/5
y = 38/5
Corner point (x, y) = (14/5, 38/5)
Finally, substitute these corner points into the objective function P = 3x + 2y to find the minimum and maximum values:
For corner point (5, 1):
P = 3(5) + 2(1)
P = 15 + 2
P = 17
For corner point (10/13, 46/13):
P = 3(10/13) + 2(46/13)
P = 30/13 + 92/13
P = 122/13
For corner point (14/5, 38/5):
P = 3(14/5) + 2(38/5)
P = 42/5 + 76/5
P = 118/5
The minimum value of P is 17, which occurs at the corner point (5, 1).
The maximum value of P is 118/5, which occurs at the corner point (14/5, 38/5).
Therefore, the minimum is P = 17 at (x, y) = (5, 1).