1. Maximize z = 16x + 8y subject to:
2x + y ≤ 30
x + 2y ≤ 24
x ≥ 0
y ≥ 0
Graph the feasibility region.
Identify all applicable corner points of the feasibility region.
Find the point(s) (x,y) that maximizes the objective function z = 16x + 8y.
The corner points are, by inspection:
(0,30),(0,12), (15,0), (24,0),
and (12, 6)[inters. of the two lines].
The points in italics do not satisfy at least one constraint.
Now evaluate the objective function at each of the feasible corner points and select the one that maximizes the objective function.
If there are two points that give the same maximum value of the objective function, then any point that lie on the line joining the two points and is located between the two points maximizes the objective function.
To graph the feasibility region, we need to plot the constraints and shade the region that satisfies all the constraints.
1. Graph the line 2x + y = 30:
- Find the x and y-intercepts by setting x = 0 and y = 0:
x = 0: 2(0) + y = 30, y = 30 → y-intercept: (0, 30)
y = 0: 2x + 0 = 30, x = 15 → x-intercept: (15, 0)
- Plot the points (0, 30) and (15, 0) and draw the line passing through them.
2. Graph the line x + 2y = 24:
- Find the x and y-intercepts by setting x = 0 and y = 0:
x = 0: 0 + 2y = 24, y = 12 → y-intercept: (0, 12)
y = 0: x + 2(0) = 24, x = 24 → x-intercept: (24, 0)
- Plot the points (0, 12) and (24, 0) and draw the line passing through them.
3. Shade the region that satisfies both constraints:
- To determine which side to shade, choose a test point not on the line.
- For example, use the origin (0, 0):
Substitute the coordinates (0, 0) into both inequalities:
2(0) + 0 ≤ 30 → 0 ≤ 30 (true)
0 + 2(0) ≤ 24 → 0 ≤ 24 (true)
- Since the origin satisfies both inequalities, shade the region below both lines.
Now we can identify the corner points of the feasibility region:
- (0, 0) is the intersection of both lines.
- (0, 12) is the intersection of the y-axis and the line x + 2y = 24.
- (15, 0) is the intersection of the x-axis and the line 2x + y = 30.
To find the point(s) that maximize the objective function z = 16x + 8y within the feasibility region, evaluate the objective function at each corner point:
- For (0, 0): z = 16(0) + 8(0) = 0
- For (0, 12): z = 16(0) + 8(12) = 96
- For (15, 0): z = 16(15) + 8(0) = 240
Thus, the point (15, 0) maximizes the objective function z = 16x + 8y within the feasibility region.