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.
If you rewrite the constraints in terms of y, for example,
2x+y≤30 as
y≤-2x + 30
then you can graph the constraints.
When y≤ something, then the feasible region is below the line, if y>0, the feasible region is above the line.
For x≥0, it is on the right of the y-axis.
What do you get for the corner points?
Once you have the corner points in the form of (x,y), you can evaluate
Z(x,y) in terms of x and y and hence compare the value of Z that maximizes its value.
To graph the feasibility region for the given constraints, we can start by graphing the lines representing the inequalities:
2x + y ≤ 30
This equation can be rewritten in slope-intercept form as y ≤ -2x + 30, which has a slope of -2 and a y-intercept of 30. We can graph this line as a solid line.
x + 2y ≤ 24
This equation can be rewritten in slope-intercept form as y ≤ -0.5x + 12, which has a slope of -0.5 and a y-intercept of 12. We can graph this line as a solid line.
Next, we need to determine the region that satisfies both inequalities. This region is the area below the line y = -2x + 30 and below the line y = -0.5x + 12. The shaded region will represent the feasible region.
To find the corner points of the feasibility region, we can solve the system of equations formed by the equations of the lines intersecting at the corners. The corner points are the points of intersection.
To maximize the objective function z = 16x + 8y, we evaluate the objective function at each of the corner points and choose the point(s) that give us the maximum value.